Krzysztof Dominik Ma´slanka
email: krzysiek2357@gmail.com

Asymptotic Properties of Stieltjes Constants

Keywords: Stieltjes constants, saddle point method, Nørlund-Rice integral Mathematicians

We present a new asymptotic formula for the Stieltjes constants which is both simpler and more accurate than several others published in the literature (see e.g. [3], [6], [13]). More importantly, it is also a good starting point for a detailed analysis of some surprising regularities in these important constants.

Introduction

The Stieltjes constants n are essentially coe¢ cients of the Laurent series expansion of the Riemann zeta function around its only simple pole at s = 1:

(s) = 1 s 1 + 1 X n=0 ( 1)
n n! n (s 1) n (1)
It is commonly believed that they are irrational numbers, and even transcendental, however no rigorous proof of this has been given [START_REF] Ma´slanka | Are the Stieltjes constants irrational? Some computer experiments[END_REF]. High precision numerical computations of them are quite a challenge (see [START_REF] Ma´slanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF] and references therein). A common and frequently cited view is that "for large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern" [14]. The …rst view is beyond any doubt, as illustrated in the Figure 1 below. In this paper, however, we will show that the second view is incorrect: not only the signs of the Stieltjes constants, but their values also show amazing regularities.

There are three asymptotic formulas for these constants in the literature ( [START_REF] Knessl | An E¤ ective Asymptotic Formula for the Stieltjes Constants[END_REF], [START_REF] Fekih-Ahmed | A New E¤ ective Asymptotic Formula for the Stieltjes Constants[END_REF], [START_REF] Bruce | An Asymptotic Expansion for the Stieltjes Constants[END_REF]). We believe that the one presented in this paper is de…nitely simpler than the others. It is also more accurate. In particular, it recreates correctly the sign of n for the particular value of n = 137 which is usually troublesome for asymptotic formulas. Most importantly, this formula can be a starting point for the analysis of the above-mentioned surprising regularities of Stieltjes constants:

n r 2 n! Re (s n ) e csn (s n ) n q n + s n + 3 2 (2)
where s n is the saddle point (see below):

s n = n + 3 2 W n+ 3 2 2 i (3)
In formula (2) c ln (2 i) is a complex constant and W is the Lambert function (sometimes called the omega function or product logarithm, see [16]). The basic tool is, as usual in such computations, the saddle point method whereas the starting point is a certain alternating sum, which, due to the still little known Nørlund-Rice formula, can be converted into an integral over the complex contour. As will be shown subsequently, global properties of this integral clearly suggest using the saddle point method.

Algorithm for calculating Stieltjes constants

This work is a natural continuation of the previous one [START_REF] Ma´slanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF]. In that work, certain numerically e¢ cient formula for Stieltjes constants was given. In the present work, we will use this formula to derive a new, e¤ective formula for asymptotics for these important constants. As it was done in [START_REF] Ma´slanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF], we will use polynomial interpolation for the (regularized) Riemann zeta function '(s):

'(s) := (s) 1 s 1 s 6 = 1 s = 1 (4)
where is the Euler constants which stems from the appropriate limit. In the mentioned interpolation, certain coe¢ cients k appear naturally, de…ned as follows:

a k (") = k X j=0 ( 1) j k j '(1 + j") (5)
where " is certain real, not necessarily small number. (In what follows we shall generally drop for simplicity this dependence in denotations: a k (") a k .) Then, after some elementary computations, we get:

n = n! " n 1 X k=n ( 1) k a k k! S (n) k = ( 1) n n! " n 1 X k=n a k k! S (n) k (6)
where S

(n) k are signed Stirling numbers of the …rst kind (see [17]). Formula ( 6) is particularly well-suited for numerical computations provided one has precomputed equidistant, high precision values of '(s) in s = 1; 1 + "; 1 + 2"; ::: (See paragraph IV of [START_REF] Ma´slanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF] for all details.) 3 Behavior of coe¢ cients a k Formula (5) has special form of an alternating sum with binomial coe¢ cients. This form suggests using the Nørlund-Rice integral which is a powerful tool for dealing with such sums. (see e.g. [START_REF] Flajolet | Mellin transforms and asymptotics: Finite di¤ erences and Rice's integrals[END_REF]).

Lemma 1 Let '(s) be holomorphic in the half-plane <(s) n 0 1 2 . Then the …nite di¤ erences of the sequence f'(k)g admit the integral representation:

n X k=n0 ( 1) k n k '(k) = ( 1) n 2 i I C '(s) n! s(s 1):::(s n) (7)
Figure 2: Rectangular contour C of integration for the right hand side of (7) for particular value k = 5 encircling points 0; 1; :::; 5. This shape is especially well suited for numerical investigations.

where the contour of integration encircles the integers fn 0 ; :::; ng in a positive direction and is contained in <(s) n 0 1 2 .

Proof. According to the Cauchy residue theorem the contour integral on the right is the sum of the residues of the integrand at s = n 0 ; :::; n which is just equal to sum on the left1 . However, before applying the above Lemma it is convenient to make several elementary transformations in [START_REF] Flajolet | On Di¤ erences of Zeta Values[END_REF].

a k = + k X j=1 ( 1) j k j '(1 + j") = = + k X j=1 ( 1) j k j (1 + j") 1 " k X j=1 ( 1)
j j k j
The last sum is

k X j=1 ( 1)
j j k j = (0) (k + 1) = H k
where (0) (s) is the polygamma function and H k P k i=1 1=i is the k th harmonic number. Finally we get:

a k = + H k " + k X j=1 ( 1) j k j (1 + j") (8)
Now choosing the rectangular contour of integration (Figure 1) and applying the above Lemma ( 7) to (5) we get:

a k = ( 1) k k! 2 i 0 @ i Z +i f k ds + k+ i Z i f k ds + k+ +i Z k+ i f k ds + +i Z k+ +i f k ds 1 A (9)
where the integrand is:

f k f k (s; ") = '(1 + s") k Y i=0 (s i)
and ' is the regularized zeta function ( 4) and is positive parameter. (Typically = 1 2 , see Fig. 2.) Deforming the rectangular contour of integration to a vertical line Re s = 1 2 and a large semicircle on the right and performing the integral along vertical line only, that is neglecting contribution from the large semicircle, which tends to zero, we get:

a k = ( 1) k k! 2 i + 1 2 i1 Z + 1 2 +i1 f k (s; ")ds
After applying functional equation for the Riemann zeta function (see e.g. [START_REF] Harold | Riemann's Zeta Function[END_REF], p. 12-16)2 :

(1 + s") = 1 2 +s" ( s" 2 ) ( 1+s" 2 ) ( s") (10)
we get

a k = + H k " + ( 1) k k! 2 i + 1 2 i1 Z + 1 2 +i1 1 2 +s" ( s" 2 ) ( 1+s" 2 ) ( s") k Y i=0 (s i) ds
Performing change of variable s ! s yields:

a k = + H k " ( 1) k k! 2 i 1 2 +i1 Z 1 2 i1 1 2 s" ( s" 2 ) ( 1 s" 2 ) (s") k Y i=0 ( s i) ds
Using elementary identity valid for integer k:

k Y i=0 ( s i) = ( 1) k k Y i=0 (s + i)
and converting the product on the right into the Pochhammer symbol usually denoted (s) n :

k Y i=0 (s + i) = (s + k + 1) (s) (s) k+1
we get:

a k = + H k " + k! 2 i 1 2 +i1 Z 1 2 i1 1 2 s" ( s" 2 ) ( 1 s" 2 ) (s) (s + k + 1) (s")ds
Now de…ning the integrand as:

f k (s; ") = 1 2 s" ( s" 2 ) ( 1 s" 2 ) (s) (s + k + 1) (s") (11)
we get

a k = + H k " + k! 2 i 1 2 +i1 Z 1 2 i1 f k (s; ")ds = (12) = + H k " + k! 2 i 0 B @ + 1 2 +i1 Z + 1 2 i1 f k (s; ")ds 2 i Res (f k (s; "); 0) 1 C A (13)
We can …nally move the line of integration from Re s = 1 2 to Re s = + 1 2 and subtract the contribution from residue of the integrand in s = 0. It turns out that this residue is:

" + H k "k! (14)
which miraculously cancels exactly the …rst and the second term in ( 12)

a k = k! 2 i + 1 2 +i1 Z + 1 2 i1 f k (s; ")ds (15)
It is convenient to introduce the following notation: The integrand in (16) has several remarkable features. It is free of singularities in the right half-plane and decays there exponentially to zero. Hence, the vertical line of integration may be freely moved to the right without any change of the integral. Therefore the integral is well-suited for applying the saddle point method. Let us now remind the following important result (see [START_REF] Erdélyi | Asymptotic Expansions[END_REF] for a very accessible presentation of this method):

g k (s; ") 1 2 s" ( s" 2 ) ( 1 s" 2 ) (s) (s + k + 1) (16)
f k (s; ") = g k (s; ") (s")
Theorem 2 The saddle-point method (or: Method of steepest descent). An integral depending of some real parameter may be approximated for large value of this parameter as

Z (x)e !(x) dx (x 0 )e !(x0) s 2 ! 00 (x 0 ) ; ! 0 (x 0 ) = 0 (17)
(The solution x 0 of the equation ! 0 (x 0 ) = 0 is the saddle point.) 3 In our case the discrete index k plays the role of parameter although it is not just multiplying factor. It is evident that in order to apply the above theorem to integral (16) one has to choose 1 and ! = log (f k (s; ")). More precisely:

! k (s; ") log (f k (s; ")) (18)
All the computations below are elementary but very tedious, so they were performed and checked with the help of Wolfram Mathematica [START_REF]Mathematica, Version[END_REF].

We shall also need the …rst and the second derivative of the integrand (16) with respect to complex variable s. Having these we can compute derivatives of ! k (s; ") as:

@ @s ! k (s; ") = @ @s f k (s; ") f k (s; ") f (1) k (s; ") f k (s; ") (19)
@ 2 @s 2 ! k (s; ") = @ 2 @s 2 f k (s; ") f k (s; ") @ @s f k (s; ") f k (s; ") ! 2 f (2) k (s; ") f k (s; ") f (1) k (s; ") f k (s; ") ! 2 (20)
Let (s) and (1) (s) denote digamma function and its …rst derivative, respectively. Introducing the following denotations:

p k (s; ") (s) (s + k + 1) + " 2 s" 2 + 1 s" 2 2 log( )
3 Historical digression. We owe the original idea of this method to Pierre Simon de Laplace (1774). Another contribution belongs to Augustin Louis Cauchy (1829). In Bernhard Riemann's unpublished notes from 1863, this method is applied to hypergeometric functions. The …nal version was published by Peter Debye (1909) who applied this method to Bessel functions. Russian historians of mathematics recently reminded contribution of Pavel Alexeevich Nekrasov, who (allegedly) discovered and used this method independently a quarter of a century before Debye. I have no opinion on this matter, since this Nekrasov was also a philosopher and used mathematics to demonstrate the necessity of the tsarist regime and the need to maintain secret services.

p

(1) k (s; ") (1) (s) (1)

(s + k + 1) + " 2 2 (1) s" 2 (1)
1 s" 2 after some elementary but tedious computations we get:

f (1)
k (s; ") @ @s f k (s; ") = g k (s; ") p k (s; ") ("s) + " 0 ("s)

In a similar way we can obtain the second derivative of f k (s; "). Introducing denotations:

q k (s; ") p k (s; ") 2 + p (1)
k (s; ") q

(1) k (s; ") 2"p k (s; ") we get:

f (2) k (s; ") @ 2 @s 2 f k (s; ") = g k (s; ") q k (s; ") ("s) + q (1)
k (s; ") 0 ("s) + " 2 00 ("s)

(22) Inserting ( 21) and ( 22) into ( 19) and (20) we get:

@ @s ! k (s; ") = p k (s; ") " 2 tan s" 2 + " 0 ("s) ("s) (23)
@ 2 @s 2 ! k (s; ") = q k (s; ") " 2 tan s" 2 2 + " 2 0 @ 00 ("s) ("s) 0 ("s) ("s) ! 2 1 A (24)
Having explicitly calculated derivatives of the integrand we are ready to apply the saddle point method (17). First we have to …nd the location of saddle points. Equating (21) or (23) to zero gives: p k (s; ") ("s) + " 0 ("s) = 0 (25)

Note that for variable s having large imaginary part we have (cf. e.g. [START_REF] Flajolet | On Di¤ erences of Zeta Values[END_REF], formula (20)):

(s) 1 (26) 0 (s) 0 00 (s) 0
The above approximations seem very radical and illegitimate, because the zeta function seems to disappear from the reasoning at this stage. Nevertheless, they are satis…ed with accuracy to many signi…cant digits along the integration path which, let us recall, can be shifted arbitrarily far to the right. After all, the zeta is present there, at least by its functional equation [START_REF] Ma´slanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF], which has been used above.

Therefore, instead of (25), we simply get:

3 + 2k + s" 2 ln 2 s" i = 0 (27)
(One has to be careful with logarithms of complex arguments so as not to ignore a case and therefore not miss a solution.) Equation ( 27) may be solved explicitly with respect to s giving the complex location of k th saddle point (for a given small parameter "):

s k = k + 3 2 "W k+ 3 2 2 i ( 28
)
where W is the Lambert function satisfying transcendental functional equation:

s = W (s)e W (s)
Incidentally, formula (28) resembles approximate formula for the imaginary parts y n of complex zeta zeros found by André LeClair (see [START_REF] Leclair | An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N[END_REF], formula ( 22)):

y n = n 11 8 W n 11 8 e
From (28) it is evident that distribution of saddle points on the complex plane scales as the inverse of parameter ".

Completion of computations

Having calculated the second derivative of the integrand and the positions of the stationary points, we can …nally use the theorem (17) and provide an asymptotic expression for a k :

a k (") ' Re k! i s 2 @ 2 @s 2 ! k (s k ; ") f k (s k ; ") ! (29)
To get the sought asymptotic formula for n coe¢ cients, all that remains is to insert (29) into the general expression [START_REF] Knessl | An E¤ ective Asymptotic Formula for the Stieltjes Constants[END_REF] and make some elementary approximations. (As always, Mathematica procedures such as Limit, Series, etc. save a lot of time and e¤ort while ensuring that the results are error free.) In particular:

@ 2 @s 2 ! k (s k ; ") ' 3 + 2k 2s 2 + " s 2 " 2 4 1 + tan s" 2 2 (30)
Using (26) we can also put:

f k (s; ") ' g k (s; ") (31)
Remembering that for large imaginary part of s we have:

(s) ' r 2 s e s s s cos(s) ' e is 2
g k (s; ") ' 1 2 1 2 s" s k 2 2s k k 2 ( s" 2 ) ( 1 s" 2 ) (32)
It is clear that, since …nally " tends to zero, it is su¢ cient to take only the …rst term in ( 6)

n ' a k (") " n " ! 0 (33)
Inserting to (33) expression for a k (") (29) together with (30), ( 31) and (32) we …nally get:

n r 2 n! Re (s n ) e csn (s n ) n q n + s n + 3 2 (
In fact, there is always a pair of mutually conjugate saddles but contributions due to their imaginary parts cancels.) It is probably quite astonishing that after making so many approximations the …nal formula for n works so well as computer experiments show convincingly. As expected, in this formula there is no longer the auxiliary parameter ", which ful…lled its important but temporary role in numerical computations (with the help of formula ( 6)), and …nally simply get shortened.

Summary of results

Let's collect the …nal results. Let c be a complex constant:

c = log(2 ) + 2 i = log(2 i)
Now asymptotics of Stieltjes constants when n ! 1 (in practice it su¢ ces that n 0) is:

n r 2 n! Re (s n ) e csn (s n ) n q n + s n + 3 2 (34)
where complex saddle points are (note that now there is no " which get shortened):

s n = n + 3 2 W n+ 3 2 2 i (35)
The 250 initial values of the complex saddles (35) are shown in the Figure 6. Very good agreement of approximated values calculated using (34) with actual values of n is shown in Figure 7. 6 Application: Signs of n As a by-product of these intricate computations, we can get a compact expression for the signs of the Stieltjes constants. Formula (34) hides the characteristic behavior of Stieltjes constants when n grows, that is large and growing oscillations with diminishing frequency superimposed on the strongly growing trend. This behavior may be demonstrated as follows. Recall higher order Stirling formula for (x):

(x) ' 1 6 r 2 e x x x 3 2 (12x + 1)
Applying it to (s n ) in (34) we get: transformations and in terms of its enormous purely numerical capabilities and …nally for its rich graphical presentations of the obtained results. Figure 9 is an example of how well Mathematica is doing to check that the contour integral ( 9) is indeed equal to the binomial alternating sum [START_REF] Leclair | An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N[END_REF]. I cannot imagine how to verify this fact with such high precision without computer support. Another example: Figure 10 shows how Mathematica solves the transcendental equations (27).

n ' 2n! Re (s n ) sn n 3

Figure 1 :

 1 Figure 1: Absolute values of 150 initial Stieltjes coe¢ cients n . Global, fast growing trend is evident. Oscillations of increasing amplitude and decreasing frequency superimposed on this trend are visible. Red dots mean positive values, blue dots mean negative values. The scale on the vertical axis is logarithmic.

Figure 3 :

 3 Figure 3: Absolute value of the integrand (16) for k = 8 and " = 25 . Vertical scale is logarithmic for better visualisation. The right half-plane of complex variable s is free of singularities. Simple poles in s = 0; 1; 2; :::; k due to factor (s) in (16) are visible.

Figure 4 :

 4 Figure 4: The logarithm of the absolute value of the integrand (11) for k = 2 and " = 2 4 . Positions of saddle points is marked by vertical lines. The saddle nature of these points is practically invisible due to the scale of the …gure. Also, three singularities for s = 0; 1; 2 merged into single peak. Better visualisation is presented on the next Figure 5.

Figure 5 :

 5 Figure 5: Typical family of fragments of absolute values of the function ! k (s; ") (18) in the vicinity of saddle points for k = 100; 101 and 102. Blue vertical segments mark the position of the saddles. The red lines are the curves of the steepest descent.

Figure 6 :

 6 Figure 6: Distribution of 250 initial saddle points on the complex plane. There are two symmetrical branches, the lower one is the complex conjugate of the upper one.

Figure 7 :

 7 Figure 7: Comparison of absolute values of actual Stieltjes coe¢ cients (green dots) with those calculated from asymptotic formula (34) (red dots) shows good agreement (except 0 ), even for that "unruly" value n = 137.

Figure 8 :

 8 Figure8: Unfortunately, the good impression after looking at Figure7diminishes a bit when we look at the graph of the ratio of the exact n values to the asymptotic values (34) for example in the range of n = 3000 3300. Although this ratio is very close to one, with an accuracy generally better than 10 4 , there are distinct, periodic structures: points are arranged on certain curves resembling the family of cotangent functions. But it is in these structures that the essence of Riemann's zeta is contained, including the Riemann hypothesis, i.e. what was rejected when the approximations (26) were made. Also note, which is somewhat surprising, that the blue points lie slightly above the red straight line that represents the value of one. But, as the saying goes, when one door shuts, another one opens. And indeed: this result opens up a whole new …eld for very fruitful research on the Stieltjes constants, which will be the subject of the next publication.

Figure 9 :

 9 Figure 9: Distribution of complex values of phase ' n given by (38) for n = 1; 2; :::; 100. It is clear that thay lie along certain smooth curve. It is also obvious that the exponent of the real part of the phase (38) controls the rapid growth of n , while the cosine of the imaginary part of the phase is responsible for the oscillations of decreasing frequency.

Figure 10 :

 10 Figure 10: Checking the Nørlund-Rice formula (7) using Mathematica.

Donald Knuth popularized this formula and attributed it to American engineer Stephen O. Rice, pioneer in the applications of probability techniques to engineering problems(1907- 1986). Knuth did it in one of the problem tasks at the end of one of the chapters of his famous work[START_REF] Knuth | The Art of Computer Programming[END_REF]. However, much earlier this formula was known to Danish mathematicianNiels Erik Nørlund (1885-1981), who included it in his extensive classic treatise[START_REF] Erik | Vorlesungen über Di¤ erenzenrechnung[END_REF]. Incidentally, the mentioned Niels Erik Nørlund was the brother of Margrethe née Norlund, later wife of the famous physicist Niels Bohr.

Such a trick to use the functional equation for the Riemann zeta function and then perform change of variable s ! s was inspired by the work[START_REF] Flajolet | On Di¤ erences of Zeta Values[END_REF], cf. equations (17), (18) and the corresponding comment.

(s n + 1 12 )e (c+1)sn q n + s n +

2(36)

For n 1 fractions 1 12 and 3 2 under the square root may be neglected since s n grows fast with n:

Applying once again Stirling formula to n! we have:

(37) Introducing …nally complex "phase" as:

(38) we get a particularly simple form of the asymptotic formula for Stieltjes constants:

Formula (39) gives almost as good approximation as (34) but it shows in a manifest way mentioned above basic properties of n (trend and oscillations). It is then clear that the statement quoted at the beginning that "Stieltjes constants [...] change signs in a complex pattern" [14] is not true. In particular, one can quickly calculate sign of n , even for extremely high n, since it is obviously equal to the sign of cos (Im ' n ) and the phase (38) can be computed e¤ectively for n at least up to 10 1;000;000 . (See [START_REF] Jasiński | The On-Line Encyclopedia of Integer Sequences[END_REF] for extensive computations of signs of Stieltjes constants using the above formulas.) For example: n sign of n 10 10 +1 10 100 +1 10 1000 +1 10 10 000 1 10 100 000 1 10 1;000 000 +1 7 Appendix -samples of Mathematica notebooks As mentioned in the main text, Wolfram's Mathematica [START_REF]Mathematica, Version[END_REF] made very tedious and convoluted computations much easier and ensured that there were no mistakes in them. This program was used very intensively -for symbolic


Ben Gikonyo

Fangbing Liu

Siddhartha De

Catherine Journet

Catherine Marichy

Alexandra Fateeva

Investigating the vapour phase synthesis of copper terephthalate Metal Organic Framework thin films by atomic/molecular layer deposition

Keywords: MOF, thin films, vapour phase synthesis, hybrid materials, atomic layer deposition, molecular layer deposition, epitaxy

Solventless synthesis and processing of Metal Organic Frameworks (MOFs) is critical to implement these materials in applied technologies. Vapour phase synthesis of MOF thin films is particularly suitable for such applications, but challenging compared to the conventional solution based methods. It is therefore compelling to advance and widen the vapour phase synthesis of MOF thin films. Crystalline copper terephthalate MOF thin films are grown in vapour phase by the means of atomic and molecular layer deposition (ALD/MLD) on different kinds of substrates. Expanding from the pioneer work, the formation of the 3D phase is clearly evidenced for the first time and the adaptability of the process to several kinds of substrates is revealed. A directional film growth is observed at the early stage of the ALD/MLD process leading to oriented MOF crystallites on a surface, when isotropical growth proceeds with the increasing number of ALD/MLD cycles. Notably, this study primary demonstrates a heteroepitaxial growth achievable in the vapour phase by using DMOF-1 single crystals as the starting surface with a lattice matching topology. Such approach offers an appealing pathway to develop MOF on MOF superlattice materials in vapour phase.

Introduction

Atomic layer deposition (ALD) is a classical thin film growth technique mainly applied to the microelectronics industry [START_REF] Plutnar | Applications of Atomic Layer Deposition in Design of Systems for Energy Conversion[END_REF][START_REF] Tripathi | Atomic Layer Deposition of P-Type Semiconducting Thin Films: A Review[END_REF][START_REF] Zhao | Atomic/Molecular Layer Deposition for Energy Storage and Conversion[END_REF][START_REF] Leskelä | Review Article: Atomic Layer Deposition of Optoelectronic Materials[END_REF] . It is based on successive self-limited reactions between gas phase precursors and a solid surface, allowing to grow a film layer by layer with an atomic scale control of the thickness [START_REF] Johnson | A Brief Review of Atomic Layer Deposition: From Fundamentals to Applications[END_REF][START_REF] George | Atomic Layer Deposition: An Overview[END_REF] . Though ALD is mostly used for the synthesis of inorganic materials, noticeable progress was made when the molecular layer deposition (MLD) of organic polymers was first reported in the 1990's [START_REF] Yoshimura | Polymer Films Formed with Monolayer Growth Steps by Molecular Layer Deposition[END_REF] , followed by the growth of hybrid materials in 2008 [START_REF] Dameron | Molecular Layer Deposition of Alucone Polymer Films Using Trimethylaluminum and Ethylene Glycol[END_REF] . The adaptability of this technique to the growth of materials using precursors that could lead to MOF was first demonstrated in 2010 by Klepper et al. who reported an ALD/MLD process of several aluminium carboxylates, although the resulting films were amorphous [START_REF] Klepper | Deposition of Thin Films of Organic-Inorganic Hybrid Materials Based on Aromatic Carboxylic Acids by Atomic Layer Deposition[END_REF] . Hence, a major breakthrough for the vapour phase synthesis of crystalline MOF thin films via ALD/MLD was performed in 2016 by Ahvenniemi et al. who reported the direct growth of crystalline copper terephthalate (CuTPA) 10 . Since this seminal study, the great majority of ALD/MLD of crystalline MOFs is reported by the group of Karppinen in Alto University 11-16 . In other cases, MOF thin films were obtained in two steps as after the ALD/MLD process, a vapour post-treatment was necessary to achieve a crystalline film [START_REF] Lausund | All-Gas-Phase Synthesis of Amino-Functionalized UiO-66 Thin Films[END_REF][START_REF] Lausund | All-Gas-Phase Synthesis of UiO-66 through Modulated Atomic Layer Deposition[END_REF][START_REF] Salmi | Studies on Atomic Layer Deposition of MOF-5 Thin Films[END_REF][START_REF] Salmi | Studies on Atomic Layer Deposition of IRMOF-8 Thin Films[END_REF][START_REF] Lausund | MOF Thin Films with Bi-Aromatic Linkers Grown by Molecular Layer Deposition[END_REF] . This later approach is less desirable, as it somehow cancels the great advantage of all-gas phase processing. Indeed, vapour phase synthesis of MOFs thin films is a highly attractive strategy when contemplating MOF integration into devices [START_REF] Stassen | Vapor-Phase Deposition and Modification of Metal-Organic Frameworks: State-of-the-Art and Future Directions[END_REF][START_REF] Su | Vapor-Phase Processing of Metal-Organic Frameworks[END_REF][START_REF] Crivello | Advanced Technologies for the Fabrication of MOF Thin Films[END_REF] . In conjunction with reducing environmental impact, solvents should be avoided for device fabrication regarding the contamination and corrosion risks [START_REF] Sánchez-Sánchez | Synthesis of Metal-Organic Frameworks in Water at Room Temperature: Salts as Linker Sources[END_REF] as well as the generated surface tension during evaporation [START_REF] Ma | Rapid Guest Exchange and Ultra-Low Surface Tension Solvents Optimize Metal-Organic Framework Activation[END_REF] . Moreover, when considering MOF chemistry, developing the vapour film growth opens great opportunity in terms of achieving superlattice structures that are especially challenging with equilibrated solution-based reactions. Therefore, ALD/MLD growth of MOF thin films represents a unique opportunity both at the fundamental level and for industrial applications. This promising technique is for now however restricted in terms of scientific groups worldwide, methods, and materials. To develop and investigate the ALD/MLD of MOFs we focused on the first reported crystalline material: CuTPA [START_REF] Ahvenniemi | Atomic/Molecular Layer Deposition: A Direct Gas-Phase Route to Crystalline Metal-Organic Framework Thin Films[END_REF] . In this reference work, the authors demonstrated that the film thickness was linear to the number of cycles, with a growth per cycle (GPC) of 3 Å at 180 °C. Grazing Incidence X-Ray Diffraction (GIXRD) data were recorded and a 2D paddle-wheel MOF-2 structure was deduced, although the experimental data were not compared against the corresponding calculated XRD diagram. No morphological characterizations by the means of electron microscopy were provided, precluding the appreciation of the surface coverage, homogeneity, and the crystals topology. Here, we present thorough characterisations of the film structure, clarify the phase identification, and inspect the morphology of the crystalline MOF films. More importantly, we extend the process by investigating the film growth on different kinds of substrates. Beyond the MOF deposition on oxide substrates, we demonstrate for the first time a heteroepitaxial MOF on MOF growth in vapour phase using DMOF-1 single crystal as the substrate and leading to an unprecedented heterostructure.

Materials and methods

A home-made manual ALD/MLD reactor was developed (see supporting information for the details, section C and Figure S9), with pressure and temperature control over the precursor cylinders and the deposition chamber. Terephthalic acid (TPA) and copper 2,2,6,6tetramethyl-3,5-heptanedione Cu(thd) 2 were used as reactants according to the reported procedure

Results and discussion

Materials deposited with an increasing number of cycles (160 and 250 cycles) were characterized by the means of scanning electron microscopy (SEM). As shown in Figure 1a and b, fairly homogeneous films with a good surface Figure 1: SEM images of the thin film obtained after 160 ALD/MLD cycles on Si (100), lower (top) and higher(bottom) magnifications (a), the film crack on the bottom image shows the oriented growth of the film. SEM images of the thin film obtained after 250 ALD/MLD cycles on Si (100), lower (top) and higher(bottom) magnifications (b). PXRD patterns of the films obtained after 160 cycles (blue) and after 250 cycles (teal), the calculated PXRD diagram for CuTPA (black), * corresponds to the free TPA (c), reaction scheme for the ALD/MLD of CuTPA (d) coverage were obtained (Figure S15a andb). The substrates were covered by well-defined rod-shaped crystallites of about 150 nm. For 160 cycles, an orientation of the crystallites perpendicular to the substrate is noticeable (Figure 1a), when increasing the number of cycles, crystallites start to grow on each other in a more isotropic way (Figure 1b). It can be noted here that the oriented growth at low number of cycles proceeds on amorphous surface of native oxide. Energy Dispersive Spectroscopy (EDS) analysis confirms the presence of Cu and C at the surface (Figure S17). Structural analysis was performed by the means of XRD. The obtained films show good crystallinity as depicted in Figure 1c. Phase identification clearly demonstrates the formation of the triclinic 3-dimensional CuTPA framework 27 , where the copper ions are solely connected by the carboxylate functions and form Cu-O chains that run in the same direction as the onedimensional pores of ca 5.2 Å (Figure 1d). This clarifies the MOF phase obtained from the ALD/MLD process, which differs from the previously assigned paddle-wheel 2-dimensional structure. The observed 3D phase is in accordance with a solventless process, as no other linker than TPA is present in the Cu coordination sphere. The diffractograms in Figure 1c show some extra Bragg peaks that have been attributed to free TPA, probably due to its low volatility that prevents the total removal during the purge step. Attempts to avoid free TPA deposition using a shorter pulse and/or a lower temperature for this precursor (140 °C instead of 180 °C) led to a considerable loss of the film crystallinity that is evidenced by a considerable decrease of the diffraction peak intensity (Figure S12). To avoid the presence of unreacted TPA, a 5 m filter gasket was implemented upstream from the reaction chamber and further studies were conducted with the standard parameters. More importantly, XRD patterns (Figure 1c) indicate that the film growth starts in an oriented manner, given the large relative intensities of the peaks corresponding to the (010) and (110) inter-reticular planes in the pattern corresponding to 160 cycles grown film. When further ALD/MLD cycles are implemented, the growth orientation becomes more random, as the relative intensities of Bragg peaks match closer to the calculated pattern for CuTPA. This points out that CuTPA growth first preferentially occurs perpendicular to the substrate along the [001] direction of the unit cell, meaning that the pores and the inorganic chains of the MOF run parallel to the substrate (Figure S10). After a certain film thickness, the ALD/MLD proceeds more isotropically. This analysis of XRD data is corroborated the SEM results (Figure 1a,b) discussed above. The reproducibility of the 3D phase film growth was checked by performing three times the deposition in the same conditions (Figure S11). MOF formation was further confirmed spectroscopically by FTIR analysis. The spectra (Figure 3 and Figure S14) show vibrations bands 1590 cm -1 and 1390 cm -1 that are assigned to the asymmetric and symmetric vibrations of the carboxylate group, respectively. The Δ between these two frequencies (Δ=200 cm -1

) is characteristic of bidentate bridged carboxylate [START_REF] Tan | Stability and Hydrolyzation of Metal Organic Frameworks with Paddle-Wheel SBUs upon Hydration[END_REF] and free TPA displays ν(C=O) and ν(C-O) bands at 1685 cm -1 and 1272 cm -1 , respectively (Figure 3). The obtained values match the ones observed for the bulk CuTPA that was synthesized in solution as a reference for this study (see SI, Figure S14a). It should be noted that the band at 1100 cm -1 , visible on the spectrum of CuTPA on Si wafer with native oxide, originates from the substrate (Figure S14b) and corresponds to the stretching band of Si-O-Si. The bulk MOF morphology can be described as a near-square shape assembly of platelets (Figure S5) while the ALD/MLD grown crystallites display a more elongated parallelepiped shape, indicative of their preferential caxis orientation.

To check the adaptability of the process and investigate the impact of the starting surface, the film growth was then performed on crystalline substrates. Sapphire with c-plane surface orientation and FTO on glass were chosen, both kinds of substrates present oxide/hydroxide functionalities at their surface that are well suited for reacting with the metallic precursor. The ALD/MLD was implemented with the same parameters as for Si (100). The SEM data evidence that a crystalline film is obtained on both substrates, but the crystallite morphology differs (Figure 2a andb). Indeed, on the sapphire, a rather compact film formed of rod-like crystallites is observed when in the case of FTO, thinner and more dispersed crystallites are deposited. For both films, a good coverage is obtained (Figure S15c andd), Cu and C are detected by EDS (Figures S18,S19). XRD analysis (Figure 2c) reveals that the same CuTPA phase is clearly formed on the sapphire substrate when in the case of FTO the obtained pattern is ill-defined and does not allow accurate phase identification. This might be due to sparser film coverage. In all cases, FTIR analysis reveals the coordination of copper by the TPA linker (Figure 3). Given the good adaptability of the process, we aimed to explore the heteroepitaxial MOF on MOF growth, using a single crystalline MOF as substrate. Assembling MOF on MOF heterostructures is of great interest to reach superstructures with tuned mechanical, adsorptive, and photophysical properties [START_REF] Haldar | Hierarchical Assemblies of Molecular Frameworks-MOF-on-MOF Epitaxial Heterostructures[END_REF] . Up to date, core-shell 30-32 , layered 33 , complex ternary [START_REF] Liu | Ternary MOF-on-MOF Heterostructures with Controllable Architectural and Compositional Complexity via Multiple Selective Assembly[END_REF] and anisotropic [START_REF] Lee | Tip-To-Middle Anisotropic MOF-On-MOF Growth with a Structural Adjustment[END_REF] MOF on MOF superstructures were reported through solutionbased synthesis. To the best of our knowledge, such superstructures are not reported through the use of the MOF growth in the vapour phase. DMOF-1 (Zn-TPA-DABCO) was chosen as the support because, in its structure, the intermetallic distances are very close to the ones in CuTPA (10.93 and 10.77 Å respectively, meaning a mismatch of ~1.5 % only, Figure 4a). Additionally, DMOF-1 is easily obtained as single crystals, offering a molecular material substrate with well-defined planar facets (Figure 4b andS8). From single crystal X-Ray diffraction analysis, the crystal orientation was determined, indicating that DMOF-1 grows along its 3 cell axes, with a axis as the preferential direction (inset Figure 4 b). The pores width of DMOF-1 is insufficient for the diffusion of the Cu(thd) 2 precursor, therefore an on-surface only ALD/MLD reaction can be expected. DMOF-1 single crystals were dispersed onto the surface of Si (100) substrate and the ALD/MLD of CuTPA was performed. From the SEM analysis of the recovered sample, DMOF-1 surface is clearly altered as it appears covered by nanometric crystallites and a film deposition on the Si wafer is observed (Figure 4b). This way, a heterostructure resembling a core-shell topology is obtained through ALD/MLD implementation on the MOF surface. More importantly, these crystallites are well-oriented perpendicular to the DMOF-1 surface all over the different facets (Figure 4b and S16), indicating that the ALD/MLD process allows directional growth on a structured three-dimensional substrate, with comparable intermetallic distances. EDS analysis indicates both the presence of Cu and Zn (Figure S20) on the crystal particles. To perform a spatial resolved analysis of the heterostructure, the CuTPA-coated crystals were transferred to an Au grid for transmission electron microscopy (TEM) analysis. The TEM image in Figure 5a shows thin crystallites sticking out of the DMOF-1 crystal. Elemental mapping was performed in STEM-EDS mode (Figure 5b andc), and clearly shows that Cu containing material is deposited on top of the Zn-based solid, demonstrating for the first time that a heterostructure is accessible using ALD/MLD on a MOF surface. In this set up, the deposition occurs both on the Si (100) and on DMOF-1 surfaces, which prevents thorough structural analysis of the heterostructure by X-Ray diffraction, nonetheless the obtained pattern displays Bragg peaks corresponding to the CuTPA phase (Figure S13).

Conclusions

In summary, this study clarifies and expands the scope of the ALD/MLD growth of crystalline MOF thin films. In particular, we demonstrate that the 3D CuTPA MOF films can be easily obtained through the vapour phase layer by layer growth and that this process is adaptable to different kinds of amorphous (Si with native oxide) or crystalline (sapphire, FTO) inorganic substrates. Importantly, for the first time an oriented MOF growth in vapour phase is detected and the orientation of the crystallites is determined. Notably, the directional MOF growth is also observed on the facets of DMOF-1 single crystals leading to an unprecedented core-shell material by vapour phase synthesis. This proves the ability of the ALD/MLD technique to accomplish heteroepitaxial growth on structured hybrid materials with lattice matching topologies. Our results indicate that the ALD/MLD deserves to be further and wider advanced for the processing of hybrid materials and the development of functional heterojunctions.

10 .

 10 kept at 180 and 110 °C, respectively. Ahvenniemi et al. reported the growth of crystalline CuTPA films between 180 °C and 195 °C, and of amorphous layers above 195 °C As no noticeable change was observed within this range, the reaction chamber temperature was set to 190 °C. Depositions were performed at a pressure of 2 mbar, using Argon as a carrier gas. First tests were performed on Si (100) substrates with the native oxide, then MOF films were grown on FTO, sapphire and DMOF-1 single crystal. The synthesis procedures and characterizations of the bulk MOF sample (Figures S3, S4 and S5), DMOF-1 single crystals (Figures S6, S7 and S8) as well as the Cu(thd) 2 precursors (Figure S1 and S2) are detailed in the supporting information (Sections A and B).

Figure 2 :

 2 Figure 2: SEM image of the thin film deposited on sapphire, a-oriented substrate (a), SEM image of the thin film deposited on FTO (b), XRD patterns of the films deposited on sapphire (teal) and on FTO (blue) both films were deposited through 250 cycles, the calculated pattern for CuTPA is shown in black (c).

Figure 3 :

 3 Figure 3: FTIR spectra of TPA (light grey), the bulk CuTPA (black), and CuTPA films grown on Si (100) with native oxide (blue), sapphire (teal) and FTO (orange).

Figure 4 :

 4 Figure 4: Representation of the DMOF-1 (left) and CuTPA(right) structures and the intermetallic distances (a), SEM images of DMOF-1 before (top left) and after 250 cycles of ALD/MLD of CuTPA.

Figure 5 :

 5 Figure 5: TEM image of DMOF-1 after 250 cycles of ALD/MLD of CuTPA (a), STEM image (b) and the corresponding elemental mapping performed by STEM-EDS (c)

Acknowledgments

The authors gratefully acknowledge Pr. Jacques Jose for the technical help, the Centre Technologique des Microstructures of the University of Lyon for providing the electron microscopy facilities, the French Ministry of National Education, Research and Technology, the University Lyon 1 and the CNRS for financial support; this work was supported by the French National Research Agency grant (ANR-17-CE09-0029-01).

Author contributions

Ben Gikonyo: investigation, validation

Conflicts of interest

The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Basil Grammaticos

Ralph Willox

Full-deautonomisation of a class of second-order mappings in ancillary form

Keywords: integrable mappings, singularity confinement, deautonomisation, integrability tests

We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E

(1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E

(1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

Introduction: singularity confinement

The term singularity confinement [START_REF] Grammaticos | Do integrable mappings have the Painlevé, property?[END_REF] describes a situation where a singularity, appearing spontaneously during the iteration of a mapping because of a particular choice of initial conditions, disappears again after a certain number of iterations. What is meant here by 'singularity' is when the inverse mapping becomes undefined, which is tantamount to the solution of the mapping losing some degree of freedom. Since it was observed that the mappings which were integrable through spectral methods do possess the singularity ]ocnmp[ B. Grammaticos and R. Willox confinement property [START_REF] Grammaticos | Singularity confinement and full-deautonomisation: a discrete integrability criterion[END_REF], it was surmised that the existence or not of the latter could offer a discrete integrability criterion.

In this sense singularity confinement is the discrete analogue of the Painlevé property which characterises differential systems that are integrable through spectral methods. (When integrability is obtained through other methods, like linearisability or even direct solvability, the relation to singularity confinement or the Painlevé property ceases, as shown in [START_REF] Ramani | Integrable systems without the Painlevé property[END_REF]). Before proceeding further let us give an example of the workings of singularity confinement in the case of a second-order mapping. We consider a mapping that belongs to Class I of the Quispel-Roberts-Thompson (QRT) family of mappings [START_REF] Quispel | Integrable mappings and soliton equations II[END_REF], according to the classification in [START_REF] Ramani | On the autonomous limit of discrete Painlevé equations[END_REF]. (An easily consultable and detailed list of the canonical forms of the QRT mappings in this classification is given in the Appendix). The mapping has the form

x n+1 + x n-1 = 1 + z x n , (1)
where z is a non-zero constant. Now suppose that, due to a specific choice of initial conditions, at some iteration m, x m takes the value 0, while x m-1 is finite and nonzero. This leads to the value x m+1 = ∞ which obviously does not depend on x m-1 and therefore corresponds to a singularity for this mapping. Iterating further, one obtains the values x m+2 = 1, x m+3 = ∞, x m+4 = 0 whereupon x m+5 is indeterminate since its value would involve ∞ -∞. The confinement of the singularity consists in the removal of this indeterminacy. To this end one invokes continuity with respect to the initial conditions, introducing a small quantity ϵ and assuming x m = ϵ instead of exactly 0. It is then straightforward to compute x m+1 , • • • , x m+4 and obtain x m+5 , whereupon taking the limit ϵ → 0 it turns out that x m+5 is no longer indeterminate and, in fact, takes exactly the value x m-1 . Thus the mapping has recovered the lost degree of freedom.

The usefulness of the singularity confinement criterion became apparent when it was combined with what is called the deautonomisation procedure [START_REF] Grammaticos | The Painlevé property -One Century later[END_REF]. The latter consists in assuming that the parameters that appear in a mapping are functions of the independent variable. Applying the singularity confinement criterion one can, in principle, obtain integrable non-autonomous extensions of a given integrable mapping. The main bulk of the discrete Painlevé equations has been discovered in this way [START_REF] Ramani | Discrete Painlevé equations: coalescences, limits and degeneracies[END_REF], [START_REF] Grammaticos | The hunting for the discrete Painlevé equations[END_REF]. In the case of mapping [START_REF] Grammaticos | Do integrable mappings have the Painlevé, property?[END_REF], we assume that z is a function of n and, again requiring that x m+5 be well defined in the sense described above, we find for z n the constraint:

z n+4 -z n+3 -z n+1 + z n = 0. ( 2
)
The solution of equation ( 2) is z n = αn + β + ϕ 3 (n) for generic values of α and β and where, for positive integers k, ϕ k (n) denotes a non-constant function with period k. The resulting non-autonomous system is a well-known [START_REF] Tokihiro | From continuous PV to discrete Painlevé equations[END_REF] discrete Painlevé equation.

Late confinement and the full-deautonomisation approach

A natural question to ask is what happens if one chooses not to implement the constraint [START_REF] Grammaticos | Singularity confinement and full-deautonomisation: a discrete integrability criterion[END_REF] for the parameter in equation (1) but, instead, pursues the iterations. In this case x m+5 takes the value ∞ and the succession of singularities becomes {0, ∞, 1, ∞, 0, ∞, 1, ∞, 0} and it turns out that x m+9 can have a finite value and can even depend on x m-1 , provided that the following constraint holds:

z n+8 -z n+7 -z n+5 + z n+4 -z n+3 -z n+1 + z n = 0. (3)
We call this situation a 'late' confinement (and in fact, infinitely many possibilities for late confinement do exist if one pursues the iterations even further). It turns out however that the solution of (3) does not have a nice secular plus periodic solution as was the case for (2): its characteristic polynomial has six complex and two real roots, which can in principle be expressed in terms of radicals, the largest real root being 1.425005268 • • • . Moreover the non-autonomous equation constructed with a z n obeying (3) is not integrable. The dynamical degree of its solution can be computed using the method introduced by Halburd [START_REF] Halburd | Diophantine integrability[END_REF], under the name of Diophantine approximation. (We remind that the dynamical degree of a rational mapping, the type of mapping we will be dealing with in this paper, is computed from the homogeneous degree d n of the iterates of the mapping, as the limit λ = lim n→∞ d 1/n n ; this limit always exists and cannot take values less than 1). The dynamical degree for this mapping, obtained numerically from the Diophantine approximation method, converges to approximately 1.42501 after 50 iterations. A first remark is that since this dynamical degree is clearly greater than 1 the degree of the successive iterates grows exponentially, a feature which indicates non-integrability for the mapping. But most important is the observation that the value of the dynamical degree coincides quite nicely with that of the largest root of the characteristic polynomial of the constraint (3). This is not a simple coincidence, as was shown in [START_REF] Mase | Deautonomisation by singularity confinement: an algebro-geometric justification[END_REF]. In that paper, the present authors in collaboration with T. Mase and A. Ramani, addressed the question of deautonomisation through an algebro-geometric approach. The main result we obtained was that when a mapping like (1) is regularised through exactly 8 blow-ups, one obtains the constraint given by the singularity confinement requirement at the first confinement opportunity. Moreover, the characteristic polynomial associated with this constraint is in fact the same as that for the isomorphism induced by the mapping on the Picard group for the surface obtained after blow-up. When more than 8 blow-ups are necessary for the regularisation of the system, the confinement constraints that are obtained give rise to non-integrable mappings, but the characteristic polynomials for the parameter constraints still coincide with those for the isomorphisms on the Picard group of the surfaces obtained through blow-up. Hence, the dynamical degrees for those mappings can be read off directly from the confinement conditions obtained from singularity confinement: they coincide with the largest roots for the characteristic polynomials for the constraints on the parameter.

The result presented above, together with the analysis of [START_REF] Mase | Deautonomisation by singularity confinement: an algebro-geometric justification[END_REF] led to the introduction of what is called the 'full-deautonomisation' method as an enhanced (with respect to singularity confinement) discrete integrability criterion. The necessity of an integrability criterion going beyond singularity confinement became apparent after the discovery of mappings which were non-integrable despite the fact that they possess only confined singularities. We know by now that infinitely many such mappings exist [START_REF] Hietarinta | Discrete Painlevé I and singularity confinement in projective space[END_REF], so let us choose an explicit example that is slightly different from the classical one given in [START_REF] Hietarinta | Singularity confinement and chaos in discrete systems[END_REF]:

x n+1 + x n-1 = x n + 1 x 2 n -1 . (4)
The singularity patterns of (4) are {±1, ∞, ∞, ∓1} which are both confining patterns, but the dynamical degree of this mapping is greater than one. Its value can be calculated

]ocnmp[ B. Grammaticos and R. Willox exactly and turns out to be equal to 3+ √ 5

2 . Deautonomising the mapping by assuming that the right-hand side is written as a n x n + b n /(x 2 n -1) and requiring that the singularity patterns remain the same, we find the constraints a n = 1 and b n+3 = b n which clearly do not have any connection to the dynamical degree of the mapping. However, as explained in [START_REF] Ramani | The redemption of singularity confinement[END_REF], the spirit of the full-deautonomisation method is to explore all possible extensions of the initial mapping which would preserve the singularity patterns. Thus we can (must) consider adding extra terms to the right-hand side of (4) which preserve the initial singularity patterns. To make a long story short: the proper extension of (4) to consider is

x n+1 + x n-1 = x n + 1 + f n x n x 2 n -1 . ( 5
)
By requiring that the singularity pattern of the original mapping be preserved, we obtain the constraint

f n+3 -2f n+2 -2f n+1 + f n = 0. ( 6
)
It can be easily shown that the largest root of the characteristic polynomial for ( 6) is

3+ √ 5
2 , i.e. exactly the dynamical degree of the mapping [START_REF] Quispel | Integrable mappings and soliton equations II[END_REF]. Several examples presented in [START_REF] Grammaticos | Singularity confinement and full-deautonomisation: a discrete integrability criterion[END_REF], [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF] have established the usefulness of the fulldeautonomisation approach as a discrete integrability criterion. Most of the examples presented in those publications concern rather simple mappings belonging to the classes I and II of the classification of the QRT canonical forms in [START_REF] Ramani | On the autonomous limit of discrete Painlevé equations[END_REF]. In the present paper we shall apply the full-deautonomisation method to a different class of mappings, class VII in [START_REF] Ramani | On the autonomous limit of discrete Painlevé equations[END_REF] (which upon integrable deautonomisation leads to discrete Painlevé equations associated to the affine Weyl group E (1) 8 , i.e. the highest one in the Sakai classification [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]), in order to show that the full-deautonomisation approach can still be made to work even for these highly non-trivial and intricate systems.

Trihomographic mappings and the ancillary representation

As we mentioned in the introduction, the singularity confinement criterion has been most useful in the derivation of discrete Painlevé equations using the deautonomisation procedure. Starting (usually) from a QRT mapping and assuming that the parameters that appear in the mapping are functions of the independent variable n, one can apply the confinement criterion in order to obtain the precise n-dependence in the parameters that ensures integrability. One difficulty lies in the initial mapping, namely: how does one choose the mapping to deautonomise in order to obtain the discrete analogue of a given Painlevé equation? What proved to be immensily useful in tackling this conundrum was the classification of the canonical forms of the QRT mappings. Nine forms were identified initially in [START_REF] Ramani | On the autonomous limit of discrete Painlevé equations[END_REF], one of which was strictly asymmetric (in the QRT terminology), but it turned out that two more forms do exist [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equation[END_REF] and which correspond to asymmetric mappings as well. (As pointed out in the introduction, a complete and detailed list of the QRT canonical forms can be found in the Appendix). Thanks to the canonical form classification it was possible to construct the discrete analogue of the Painlevé VI equation [START_REF] Grammaticos | On a q-discrete analogue of the Painlev 'e VI equation[END_REF], in QRT-symmetric form, deautonomising a mapping belonging to the canonical class VI (the numbering here is purely coincidental). It was thus expected that mappings from the classes VII and VIII would give rise to discrete Painlevé equations associated to the affine Weyl group E

(1) 8 of the Sakai classification [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]. In particular, class VII mappings would give rise to additive equations while class VIII ones would correspond to multiplicative ones. (Unfortunately no canonical form could be proposed which upon deautonomisation would lead to the elliptic discrete Painlevé equations that were discovered by Sakai). While the approach sketched in the introduction appears straightforward, there exist technical difficulties which make the application of deautonomisation quite arduous. For example, starting form an autonomous expression such as

(x n+1 -x n -z 2 )(x n-1 -x n -z 2 ) + 4z 2 x n x n+1 -2x n + x n-1 -2z 2 = f (x n ), (7)
it is a priori not clear at all by which combinations of z n+1 , z n and z n-1 the occurences of the parameter z should be replaced to make the calculations involved in the deautonomisation tractable. Thus the derivation of E

8 -associated discrete Painlevé equations had to wait until their explicit geometrical construction [START_REF] Ohta | An affine Weyl group approach to the 8-parameter discrete Painlevé equation[END_REF], when it became clear that the generic symmetric, additive, equation has the form

(x n -x n+1 +(z n +z n+1 ) 2 )(x n -x n-1 +(z n +z n-1 ) 2 )+4x n (z n +z n+1 )(z n +z n-1 ) (z n +z n-1 )(x n -x n+1 +(z n +z n+1 ) 2 )+(z n +z n+1 )(x n -x n-1 +(z n +z n-1 ) 2 ) = R(x n ), ( 8
) with R given by R(x n ) = 2 x 4 n + S 2 x 3 n + S 4 x 2 n + S 6 x n + S 8 S 1 x 3 n + S 3 x 2 n + S 5 x n + S 7 , (9)
where S k (k ≤ 8) is the elementary symmetric function of degree k in the 8 variables z n + κ i n (i = 1, . . . , 8), and where the parameters κ i are, generically, functions of the independent variable as well.

Another important milestone in the solution of this problem was the derivation in [START_REF] Ohta | An affine Weyl group approach to the 8-parameter discrete Painlevé equation[END_REF] of the basic Miura transformations in E

(1) 8 , the form of which actually turned out to be suitable for the representation of all discrete Painlevé equations [START_REF] Grammaticos | On a novel representation of discrete Painlevé equations[END_REF]. This form was dubbed 'trihomographic' and in the simplest, additive and symmetric, case it has the form

x n+1 -(z n + z n-1 + k n ) 2 x n+1 -(z n + z n-1 -k n ) 2 x n-1 -(z n + z n+1 + k n ) 2 x n-1 -(z n + z n+1 -k n ) 2 x n -(2z n + z n-1 + z n+1 -k n ) 2 x n -(2z n + z n-1 + z n+1 + k n ) 2 = 1. ( 10
) Most importantly, it turned out that the trihomographic form ( 10) is, in fact, strictly equivalent to a mapping of the form (8) with a right-hand side given by

R(x n ) = x n -k 2 n 2z n + z n-1 + z n+1 + 2z n + z n-1 + z n+1 . (11)
Note however that this result does not cover the case of general additive E

8 -associated equations since, as explained in [START_REF] Grammaticos | On a novel representation of discrete Painlevé equations[END_REF], these can only be expressed in terms of a system of four coupled trihomographic equations. While this is perfectly acceptable in principle, it is not very convenient from a practical (computational) point of view, when one would like to apply the deautonomisation procedure. Fortunately a solution to this final hurdle also exists.

]ocnmp[ B. Grammaticos and R. Willox It was observed in [START_REF] Ramani | Singularity analysis for difference Painlevé equations associated with the affine Weyl group E8[END_REF] that by introducing an ancillary variable ξ as

x n = ξ 2 n , (12)
it is possible to write the right-hand side (9) as

R(x n ) = 2ξ n Π(ξ n ) + Π(-ξ n ) Π(ξ n ) -Π(-ξ n ) , (13)
where Π(ξ) is given by

Π(ξ n ) = 8 i=1 (z n + κ i n + ξ n ). ( 14
)
It is then straightforward to show that equation ( 8) with r.h.s. ( 9) can be written as

x n+1 -(ξ n -z n -z n+1 ) 2 x n+1 -(ξ n + z n + z n+1 ) 2 x n-1 -(ξ n -z n -z n-1 ) 2 x n-1 -(ξ n + z n + z n-1 ) 2 = 8 i=1 (κ i n + z n -ξ n ) 8 i=1 (κ i n + z n + ξ n ) . ( 15
)
As we will see in Section 4, this factorised 'ancillary' form -obtained at the price of introducing the ancillary variable ξ -greatly simplifies the application of the singularity confinement criterion. It is also important to point out here that, thanks to the introduction of the appropriate ancillary variable it was possible to propose what is probably the simplest form [START_REF] Grammaticos | From trihomographic to elliptic Painlevé equations[END_REF] of the discrete elliptic Painlevé equation. Furthermore, when in Section 5 we shall use the full-deautonomisation method to study general confining, additive, mappings with the same left-hand sides as the class VII mappings (i.e. equations of the form [START_REF] Ramani | Discrete Painlevé equations: coalescences, limits and degeneracies[END_REF] with general right-hand sides but which have the singularity confinement property) it will become clear that the ancillary forms of these mappings are in fact a crucial and necessary ingredient, without which there would be no hope at all of being able to implement it.

Singularity confinement in the ancillary representation

A first type of singularity of equation ( 15) is obvious: x n+1 does not depend on the value of x n-1 (for generic x n-1 ) when the right-hand side of ( 15) is equal to zero or infinity. Given the structure of [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF] the difference between these two cases corresponds to a simple change of sign in ξ n which, however, is a change that leaves the equation invariant and it therefore suffices to study just one of these two possibilities. Here we shall choose the case in which we encounter a zero in the right-hand side of [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF]. So let us assume that at some iteration m we have ξ m = κ j m +z m , for some specific j. Balancing this with the left-hand side we find x m+1 = (κ j mz m+1 ) 2 and taking the square root we find ξ m+1 = ±(κ j mz m+1 ). Iterating further we obtain, in the left-hand side, a rational factor containing x m+2 multiplied by the expression

x m -(ξ m+1 -z m+1 -z m ) 2 x m -(ξ m+1 + z m+1 + z m ) 2 ≡ (κ j m + z m ) 2 -(ξ m+1 -z m+1 -z m ) 2 (κ j m + z m ) 2 -(ξ m+1 + z m+1 + z m ) 2
, which is either zero or infinity depending on the choice of sign in ξ m+1 . In either case it must be balanced by a zero factor or an infinity on the right-hand side of the equation if we want to have an opportunity for the singularity to be confined at this stage. This balance can of course be achieved using any of the eight factors in the right-hand side of (15) at n = m + 1 but, for the sake of simplicity, let us decide that it occurs in the factor that contains κ j m+1 . This factor takes the form

κ j m+1 + z m+1 -ξ m+1 κ j m+1 + z m+1 + ξ m+1 ≡ κ j m+1 + z m+1 ∓ (κ j m -z m+1 ) κ j m+1 + z m+1 ± (κ j m -z m+1 )
.

It is straightforward to show that whatever the choice of sign for ξ m+1 the condition for the two sides of (15) to balance (either through the numerator or the denominator) is always

κ j m+1 + κ j m = 0. ( 16
)
Repeating this analysis after introducing a small parameter ϵ, just as we did for (1), shows that ξ m+2 indeed depends on x n-1 if we require κ j to obey the constraint [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], and therefore that with this choice of κ j this particular singularity is confined immediately after it occurs. As a result, since there are eight ways in which this type of singularity can arise, we have to have all eight κ i change sign at each iteration for the singularity to be confined in all possible cases:

∀ i = 1, . . . , 8 : κ i n = κ i 0 (-1) n . (17)
However, the singularity examined above is not the only possible one for [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF]. Another singularity might arise when ξ m becomes infinite but since the ancillary variable ξ is present on both sides of the equation, one can in fact make sure that these infinities balance out precisely. This is tantamount to infinity not becoming a singularity for [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF], meaning that x m+1 does indeed depend on x m-1 . For this to happen the following condition for the parameter z must be satisfied:

z m+1 -2z m + z m-1 = 1 2 8 i=1 κ i m . (18)
Since only the combinations z n + z n+1 and z n + z n-1 appear in the left-hand side of [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF], it is possible to introduce a gauge z n → z n + (-1) n δ on z n , with δ = -1 8 8 i=1 κ i 0 , which leaves the equation invariant if we redefine the κ i as κ i n → κ i n -(-1) n δ, thereby cancelling the right-hand side in [START_REF] Grammaticos | On a q-discrete analogue of the Painlev 'e VI equation[END_REF] if all κ i are of the form [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equation[END_REF]. We are thus left with the constraint

z n+1 -2z n + z n-1 = 0, ( 19
)
the solution of which is z n = αn + β, which together with [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equation[END_REF] under the condition that

8 i=1 κ i 0 = 0, (20)
ensures that equation [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF] does not possess a singularity at infinity and that all other singularities are confined as soon as they arise. The equation thus obtained corresponds to the additive E ]ocnmp[

B. Grammaticos and R. Willox

A few comments are in order here. First of all, given the form of equation ( 15) one might have the impression that ξ m = 0 also corresponds to a singularity. A straightforward calculation however shows that this is not the case: x m+1 still depends on x m-1 even when ξ m = 0. This is also easily assessed on the equivalent form [START_REF] Grammaticos | The hunting for the discrete Painlevé equations[END_REF] with right-hand side as in [START_REF] Tokihiro | From continuous PV to discrete Painlevé equations[END_REF], for which x n = 0 is clearly not a singularity. Furthermore, given the relation of the ancillary equation [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF] to the non-ancillary form ( 8)- [START_REF] Tokihiro | From continuous PV to discrete Painlevé equations[END_REF], it is natural to wonder whether the singularity structures we found in the case of the ancillary representation are exactly the same in the non-ancillary one. While this is easily checked for the case where x m = ξ 2 m = ∞, for which a straightforward calculation yields condition [START_REF] Grammaticos | On a q-discrete analogue of the Painlev 'e VI equation[END_REF] if one requires that the equation in non-ancillary form does not possess a singularity at x m = ∞, this is not at all obvious for the singularities at finite positions. Although, with sufficient hindsight, it can be checked that x m = (κ j m + z m ) 2 is indeed a singularity of ( 8)-( 9) and that the conditions [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equation[END_REF] ensure that the ensuing singularity patterns are the same as for the ancillary case, this seems to be very difficult to ascertain directly on (8)- [START_REF] Tokihiro | From continuous PV to discrete Painlevé equations[END_REF]. While finding the singularities is certainly possible, performing the singularity confinement analysis quickly leads to highly intractable calculations. This is the reason why the ancillary form is in our opinion crucial in the singularity analysis of equations such as [START_REF] Grammaticos | The hunting for the discrete Painlevé equations[END_REF] or more generally [START_REF] Ramani | Discrete Painlevé equations: coalescences, limits and degeneracies[END_REF].

A last comment concerns the type of singularity patterns that we have investigated here. These involved clearly very special choices of confinement (or even 'non-singularity' for the behaviour at infinity). While it is certainly possible to investigate other, more general, singularity patterns that involve longer singularity patterns and/or confinement conditions that mix different κ i , such analysis quickly becomes quite lengthy and complicated. Faced with the profusion of the possible singular behaviours we decided to focus on a situation where a single singularity would control integrability. In the following we shall therefore limit ourselves, for singularities at finite positions, to the study of 'minimal' length singularity patterns such as those above: entering and exiting immediately, at the same iteration. However, for the singularity that might arise at ξ m = ∞ we shall allow for singularity patterns of arbitrary length.

A general class VII mapping and its integrability properties

Having set the frame, we now proceed to examine the confinement structure and possible integrability properties of a class VII mapping where the right-hand side can be a rational expression involving polynomials of degree higher (but also lower) than in [START_REF] Tokihiro | From continuous PV to discrete Painlevé equations[END_REF]. As explained above, since we want to be able to ensure confinement for the singularities of the mapping in the easiest way possible (at least for singularities that arise at finite positions) we are going to work within the ancillary representation and thus we will consider equations of the form

x n+1 -(ξ n -z n -z n+1 ) 2 x n+1 -(ξ n + z n + z n+1 ) 2 x n-1 -(ξ n -z n -z n-1 ) 2 x n-1 -(ξ n + z n + z n-1 ) 2 = k i=1 (κ i n + z n -ξ n ) k i=1 (κ i n + z n + ξ n ) . ( 21
)
where k is a positive integer, not necessarily equal to 8. The general, non-ancillary, form of these mappings is similar to that of (8),

(x n -x n+1 +(z n +z n+1 ) 2 )(x n -x n-1 +(z n +z n-1 ) 2 )+4x n (z n +z n+1 )(z n +z n-1 ) (z n +z n-1 )(x n -x n+1 +(z n +z n+1 ) 2 )+(z n +z n+1 )(x n -x n-1 +(z n +z n-1 ) 2 ) = R k (x n ), (22)
but now with a right-hand side that, depending on the parity of k, takes the form

∀ k odd : R k (x) = 2 e k 1 x k-1 2 + e k 3 x k-3 2 + • • • + e k k-2 x 1 + e k k e k 0 x k-1 2 + e k 2 x k-3 2 + • • • + e k k-3 x 1 + e k k-1 , (23)
or

∀ k even : R k (x) = 2 e k 0 x k 2 + e k 2 x k-2 2 + • • • + e k k-2 x 1 + e k k e k 1 x k-2 2 + e k 3 x k-4 2 + • • • + e k k-1 , (24)
where e k m denotes the degree m elementary symmetric polynomial in k variables

z n +κ i n (i = 1, . . . , k), e k m = 1≤j 1 <•••<jm≤k (z n + κ j 1 n ) • • • (z n + κ jm n ), (25)
and where we define e k 0 to be identically 1. All the mappings defined by equation ( 21), for general k, possess singularities at ξ m = ±(κ j m + z m ) (i = 1, . . . , k) and it should be clear from the analysis carried out in section 4 that the confinement of these singularities works in exactly the same way as in the special case k = 8: requiring that all parameters κ i n have a period 2 dependence in the independent variable n,

∀ i = 1, . . . , k : κ i n = κ i 0 (-1) n , (26)
ensures that all singularities that arise at ξ m = ±(κ j m + z m ) are immediately confined. This then only leaves the behaviour at ξ m = ∞ to be considered in detail.

Expanding both sides of equation ( 21) in powers of 1 ξn we obtain

1 - 4 ξ n (z n+1 + 2z n + z n-1 ) + o( 1 ξ n ) = (-1) k - 2(-1) k ξ n kz n + k i=1 κ j n + o( 1 ξ n ), (27)
which clearly cannot hold at the limit ξ n → ∞ when k is odd. Hence, for odd values of k, equation ( 21) necessarily always has a singularity at ξ n = ∞. However, when k is even, the condition

z n+1 + (2 - k 2 )z n + z n-1 = 1 2 k i=1 κ i n , (28)
ensures that the equation does not become singular at ξ n = ∞. As before, a suitable gauge on the parameters,

z n → z n + (-1) n δ k , κ i n → κ i n -(-1) n δ k with δ k = -1 k k
i=1 κ i 0 , leaves the equation invariant and allows one to put the sum of all κ i equal to zero for κ i of the form [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF]. In this particular gauge we are then left with the constraints

z n+1 + (2 - k 2 )z n + z n-1 = 0. ( 29
)
]ocnmp[ B. Grammaticos and R. Willox and

k i=1 κ i 0 = 0. ( 30
)
We remark that for k = 8 we find, as expected, the conditions ( 19) and [START_REF] Grammaticos | On a novel representation of discrete Painlevé equations[END_REF].

Next we turn to the case where infinity is indeed a singularity. Let us first, as an example, consider the case where this singularity confines after one step, leading to the pattern {∞, ∞}. It turns out that this pattern can be confined both for even as well as for odd k. Starting from a value ξ m = 1/ϵ, we find that ξ m+1 ∼ 1/ϵ as well but that ξ m+2 can take finite values that depend on x m-1 when

z m+3 + 2z m+2 + 2z m+1 + z m = k 2 (z m+2 + z m+1 ) + 1 2 k i=1 (κ i n+1 + κ i n+2 ), (31)
for even k and

z m+3 + 2z m+2 + 2z m+1 + z m = k + 1 2 (z m+2 + z m+1 ) + 1 2 k i=1 (κ i n+1 + κ i n+2 ), (32)
for odd k. Note that for κ i that satisfy the confinement conditions ( 26), the sums k i=1 (κ i n+1 + κ i n+2 ) in the right-hand sides of these equations are always zero, irrespective of the gauge one chooses for the parameters, and the two equations can thus be cast in the form,

z m+3 + 2 -(k + 1)/2 (z m+2 + z m+1 ) + z m = 0, (33)
for all integers k ≥ 1, where (k + 1)/2 stands for the integer part of the fraction k+1 2 . We shall not examine any more special cases and immediately give the confinement condition for the general case where the singularity pattern at infinity consists in a succession of infinities {∞, ∞, • • • , ∞} of length ℓ + 1, which for ℓ = 0 includes the case where ξ = ∞ is not a genuine singularity of the equation, in case k is even. In fact, it turns out that a pattern corresponding to an even value for ℓ can only exist in case k is even as well. Singularity patterns with odd ℓ on the other hand exist for both even as well as for odd values of k.

The general confinement condition for a singularity pattern {∞, ∞, • • • , ∞} of length ℓ + 1 is

z m+ℓ+2 + 2 -(k + 1)/2 (z m+ℓ+1 + • • • + z m+1 ) + z m = 1 2 k i=1 κ i m+1 + • • • + κ i m+ℓ+1 ,
(34) for any non-negative value of ℓ if k is even, or only for odd ℓ when k is odd. As before, when ℓ is odd the right-hand side in this constraint is automatically zero when the κ i satisfy the confinement conditions [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF]. However, for even values of ℓ the right-hand side of the constraint can still be put to zero in the aforementioned gauge

z n → z n + (-1) n δ k , κ i n → κ i n -(-1) n δ k with δ k = -1 k k i=1 κ i 0 .
Hence, in this gauge the confinement conditions can be summarized as

z m+ℓ+2 + 2 -(k + 1)/2 (z m+ℓ+1 + • • • + z m+1 ) + z m = 0, ( 35
)
for odd k and ℓ, or for any non-negative ℓ if k is even, and where the bracket notation is again used to indicate the integer part of the argument. Under the hypothesis that the full-deautonomisation method also works for mappings of this particular class, the largest root of the characteristic equation associated to the constraint (35),

λ ℓ+2 + 2 -(k + 1)/2 (λ ℓ+1 + • • • + λ) + 1 = 0, ( 36
)
should coincide with the dynamical degree of the mapping. It is easy to verify that whenever the inequality

2 + 2 ℓ + 1 = 4 - 2ℓ ℓ + 1 < k + 1 2 (37)
is satisfied, the characteristic polynomial (36) must have a root that is greater than 1 and hence, under the assumption that the full-deautonomisation method indeed works, the corresponding mapping should be non-integrable. The inequality (37) is always satisfied when k ≥ 9, for any value of ℓ, and all corresponding mappings should have dynamical degrees greater than 1. On the other hand, inequality (37) can never be satisfied for k = 1, 2, 3 or 4, whatever ℓ one chooses, and the resulting mappings should be integrable regardless of the length of the singularity pattern at ∞. These results are summarized in Table 1, in which the largest roots of the characteristic polynomial (36) are listed for the basic cases: white squares correspond to integrable mappings, grey ones to non-integrable mappings and black squares indicate impossible combinations of k and ℓ.

`k 1 2 3 4 5 6 7 8 9 Interpreting the values in Table 1 as the dynamical degrees of the mappings that correspond to those particular combinations of k and ℓ, it is clear that the mappings for k = 1, 2, 3 and 4 must be very special since they should be integrable for any length of the singularity pattern at infinity, and hence also when that singularity does not confine at all. This is only possible if the corresponding mappings are linearisable. It turns out ]ocnmp[ B. Grammaticos and R. Willox that this is indeed the case, but also that the cases k = 1, 2 and k = 3, 4 are in fact quite different.

• • • • • • • • • • • • • • • • • • 0 1 2
For k = 1 we have

(x n -x n+1 + (z n + z n+1 ) 2 )(x n -x n-1 + (z n + z n-1 ) 2 ) + 4x n (z n + z n+1 )(z n + z n-1 ) (z n + z n-1 )(x n -x n+1 + (z n + z n+1 ) 2 ) + (z n + z n+1 )(x n -x n-1 + (z n + z n-1 ) 2 ) = 2(z n + κ 1 n ), (38)
a mapping which turns out to have bounded degree growth when κ 1 obeys ( 26), for arbitrary z n . As was shown in [START_REF] Tamizhmani | On the limits of discrete Painlevé equations associated to the affine Weyl group E8[END_REF], under the constraint [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF], this equation, is just one obtained in [START_REF] Ramani | Linearisable QRT mappings[END_REF] where it was shown that it can be transformed to the Gambier-type [START_REF] Grammaticos | The Gambier Mapping revisited[END_REF] mapping (y n + y n+1 )(y n + y n-1 ) = f n y n , where f n is a free fuction of n. Moreover, at x = ∞ it has an 'anticonfined' singularity rather than a genuinely unconfined one:

. . . , ∞, ∞, ∞, h(x 0 ), x 0 , ∞, ∞, ∞, . . . , (39)
where the function h(x 0 ) is obtained from the inverse map of (38) from the initial conditions (x 0 , x 1 = ∞). Note that such anticonfined singularities are perfectly compatible with the linearisable and integrable character of these mappings [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF]. Furthermore, it is known that mappings with bounded degree growth are either periodic or can be transformed into projective mappings on P 2 (C) [START_REF] Blanc | Degree growth of birational maps of the plane[END_REF]. In fact, when z n is not free but obeys the constraint

z n+ℓ+2 + z n+ℓ+1 + • • • + z n+1 + z n = 0, (40)
i.e. when the singularity at infinity in (38) indeed confines after ℓ steps (ℓ odd), the mapping becomes periodic with period ℓ + 3 (which is the same period as for the z n ). The same result holds for the mapping at k = 2, for which we find the equation:

(x n -x n+1 + (z n + z n+1 ) 2 )(x n -x n-1 + (z n + z n-1 ) 2 ) + 4x n (z n + z n+1 )(z n + z n-1 ) (z n + z n-1 )(x n -x n+1 + (z n + z n+1 ) 2 ) + (z n + z n+1 )(x n -x n-1 + (z n + z n-1 ) 2 ) = z n + x n -(κ 1 0 ) 2 z n , (41)
which has bounded degree growth as well for arbitrary z n , when the κ i satisfy [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF]. Equation ( 41) is also linearisable: it was first identified in [START_REF] Grammaticos | Discretising the Painlevé equations à la Hirota-Mickens[END_REF] where it was shown that it can be integrated through a transformation to the "standard" Gambier equation

(y n + y n+1 )(y n + y n-1 ) = f n (y 2 n -1)
, where f n is again a free function of n. As for the case k = 1, when z n is indeed a free function of n, this mapping also has an anticonfined singularity at x = ∞ of the form (39). On the other hand, when z n obeys the confinement constraint (40) (but now for any non-negative integer ℓ) the mapping becomes periodic with period ℓ + 3.

The mappings obtained at k = 3 and k = 4 on the contrary have linear (unbounded) degree growth and possess a genuine unconfined singularity at x = ∞ which, however, is still compatible with their linearisable and therefore integrable character.

For k = 3 we find

(x n -x n+1 + (z n + z n+1 ) 2 )(x n -x n-1 + (z n + z n-1 ) 2 ) + 4x n (z n + z n+1 )(z n + z n-1 ) (z n + z n-1 )(x n -x n+1 + (z n + z n+1 ) 2 ) + (z n + z n+1 )(x n -x n-1 + (z n + z n-1 ) 2 ) = 2 e 3 1 x n + e 3 3 x n + e 3 2 . (42)
When the κ i (i = 1, 2, 3) satisfy [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF] this is a linearisable mapping of what has been called the third kind, the integration of which was presented in [START_REF] Tamizhmani | On the limits of discrete Painlevé equations associated to the affine Weyl group E8[END_REF]. When the z n do obey the confinement condition at x = ∞,

z n+ℓ+2 + z n = 0, (43)
for odd ℓ, the mapping becomes periodic with period 2(ℓ + 2), i.e. the periodicity of the z n .

For k = 4 we find

(x n -x n+1 + (z n + z n+1 ) 2 )(x n -x n-1 + (z n + z n-1 ) 2 ) + 4x n (z n + z n+1 )(z n + z n-1 ) (z n + z n-1 )(x n -x n+1 + (z n + z n+1 ) 2 ) + (z n + z n+1 )(x n -x n-1 + (z n + z n-1 ) 2 ) = 2 x 2 n + e 4 2 x n + e 4 4 e 4 1 x n + e 4 3 , (44)
which is linearisable for κ i that obey the constraint [START_REF] Mase | Integrable mappings and the notion of anticonfinement[END_REF]. Equation ( 44) was first identified in [START_REF] Ramani | Linearisable QRT mappings[END_REF] and [START_REF] Grammaticos | Discretising the Painlevé equations à la Hirota-Mickens[END_REF] and its detailed integration was given in [START_REF] Ramani | Higher linearisable mappings and their explicit integration[END_REF] (equation (83) in that reference). When the z n satisfy the condition (43) the mapping becomes periodic with period 2(ℓ + 2), for any non-negative integer value for ℓ.

The cases k = 5, 6 are more interesting. For both cases the constraint on z n , for the singularity at infinity to confine, reads

z n+ℓ+2 -(z n+ℓ+1 + • • • + z n+1 ) + z n = 0, (45)
for odd ℓ in the case k = 5 and for general non-negative integer ℓ for k = 6, in the gauge [START_REF] Halburd | Elementary exact calculations of degree growth and entropy for discrete equations[END_REF]. For ℓ = 1 this leads to a solution z n = αn + β + γ(-1) n and the corresponding equations are discrete Painlevé equations associated with the affine Weyl groups E

(

x n -x n+1 + (z n + z n+1 ) 2 )(x n -x n-1 + (z n + z n-1 ) 2 ) + 4x n (z n + z n+1 )(z n + z n-1 ) (z n + z n-1 )(x n -x n+1 + (z n + z n+1 ) 2 ) + (z n + z n+1 )(x n -x n-1 + (z n + z n-1 ) 2 )
= 2 e 5 1 x 2 n + e 5 3 x n + e 5 5

x 2 n + e 5 2 x n + e 5 4 , (46)
for k = 5 and E

(

x n -x n+1 + (z n + z n+1 ) 2 )(x n -x n-1 + (z n + z n-1 ) 2 ) + 4x n (z n + z n+1 )(z n + z n-1 ) (z n + z n-1 )(x n -x n+1 + (z n + z n+1 ) 2 ) + (z n + z n+1 )(x n -x n-1 + (z n + z n-1 ) 2 ) = 2 x 3 n +
for k = 6 respectively. Both equations were first derived in [START_REF] Tamizhmani | On the limits of discrete Painlevé equations associated to the affine Weyl group E8[END_REF].

For k = 6 and ℓ = 0 on the other hand, z n must satisfy z n+2z n+1 + z n = 0 and is therefore expressed in terms of the cubic roots of unity. The corresponding equation is of the form (47), but where we must impose 6 i=1 κ i = 0, and the mapping turns out to be periodic with period 6.

]ocnmp[ B. Grammaticos and R. Willox When ℓ > 1, for the cases listed in Table 1 for k = 5 and k = 6, we verified that numerical estimates of the dynamical degree given by Halburd's Diophantine method converge to the largest root of the characteristic polynomial for condition (45), thus showing that our full-deautonomisation approach indeed yields the correct value for the dynamical degree for those mappings.

The same holds for the cases k = 7 and k = 8 when ℓ ≥ 1. The confinement condition on z n is

z n+ℓ+2 -2(z n+ℓ+1 + • • • + z n+1 ) + z n = 0, (48)
which always has a root greater than 1 if ℓ ≥ 1. Note that for k = 8, ℓ = 0 however, z n takes the form z n = αn + β and the equation one obtains is of course nothing but the generic additive E

(1)

8 -associated discrete Painlevé equation ( 8)-( 9) encountered in the previous subsection.

Numerical calculations for the dynamical degrees of the mappings with k = 7, 8 and 9 for ℓ ≥ 1 again converge to the values indicated in Table 1, thus vindicating the fulldeautonomisation method even for such highly complicated mappings. Since all the values for the dynamical degrees thus obtained are greater than 1, the corresponding mappings are all non-integrable. In fact, for k > 8 the equations obtained with z n given by (47) are always non-integrable. We have computed the dynamical degree of the first few cases, up to k = 16 and the computed degrees are in perfect agreement with the value of the largest root of the characteristic polynomial (36).

Note that among all equations ( 21) that have confining singularities, only three have autonomous limits: the only cases in which z n = constant (combined with κ i being zero) is a solution to (35) are the cases of equations ( 46), ( 47) and ( 8)- [START_REF] Tokihiro | From continuous PV to discrete Painlevé equations[END_REF]. All other equations necessarily require a non-trivial n-dependence in z n in order to confine. However, one can consider the case of an autonomous mapping where z n , being constant, cannot satisfy the condition (35) and hence, for which the singularity at x = ∞ is unconfined when k = 7 or k ≥ 9. In this case the singularity pattern corresponds to the unconfined sequence {∞, ∞, • • • }, i.e. the limit ℓ → ∞ of the confined pattern. In this limit, the largest root of (36) converges to a value that can be computed exactly. It suffices to rewrite (36) as

1 + 2 -(k + 1)/2 1 -λ -ℓ-1 λ -1 + λ -ℓ-2 = 0
and take the limit ℓ → ∞ supposing that there exists a λ > 1, which is true for k ≥ 5. This yields

λ = k + 1 2 -1, (49)
which is precisely the value for the dynamical degree obtained for all mappings with k ≥ 5 that do not confine at infinity, and for all autonomous mappings at k = 7 and k ≥ 9 in particular.

Conclusions

In this paper we have combined the full-deautonomisation approach with the ancillary representation of mappings. The latter was introduced in order to palliate a difficulty encountered in the derivation of discrete Painlevé equations by deautonomising QRT mappings. All QRT mappings associated with classes II to VI have a factorised left-hand side of the form F (x n+1 , x n )F (x n-1 , x n ), while class I is even simpler. Studying the behaviour of their singularities is therefore straightforward. However this is not the case with class VII and VIII mappings. Here the study of the singularity is complicated and, in practice, prohibitively so. The introduction of the ancillary representation, which consists in just replacing the dependent variable by some more convenient one solves this problem, since it provides a factorised form of the equation. Moreover it obviates the distinction between classs VII and VIII since the only difference between the two is a different choice of ancillary variable: instead of the relation x = ξ 2 for additive, class VII, equations we have x = ξ + 1/ξ for the multiplicative, class VIII, ones. And, as a bonus, the ancillary representation, through the relation x = θ 2 1 (ξ)/θ 2 0 (ξ), where the θ are theta functions, offers a natural way to represent the elliptic discrete Painlevé equations [START_REF] Grammaticos | From trihomographic to elliptic Painlevé equations[END_REF]. With the introduction of this new parametrisation a host of new, E

8 associated discrete Painlevé equations, were derived.

The full-deautonomisation approach was introduced in order to palliate another difficulty. While all discrete equations integrable through spectral methods possess confined singularities, the singularity confinement property is not sufficient for the integrability of rational mappings. The latter is associated to the low growth of some characteristic (typically the degree growth of some initial condition) and while singularity confinement is associated with factorisations and simplifications that do lower the degree, it turns out that in some cases this does not suffice in order to curb the exponential degree growth. To this end the full-deautonomisation approach was introduced, requiring confinement of the singularities of the most general non-autonomous form. (Sometimes this necessitates the extension of the mapping by introducing new terms which do not alter the confining singularity pattern, but finding the proper terms to add is often non trivial). As was shown in [START_REF] Mase | Deautonomisation by singularity confinement: an algebro-geometric justification[END_REF] the deautonomisation constraints allow one to calculate the dynamical degree of the mapping, thus making it possible not only to ascertain the integrable character but, in the presence of non-integrability, to have a precise knowledge of the growth properties of the system.

Since the first confining non-integrable systems that were discovered all took the form of QRT class I mappings, it was natural that most applications of the full-deautomisation method to date had to do with systems belonging to that class (and some of class II). In this paper we have radically changed our perspective and decided to investigate the validity of the method in the case of the mappings from class VII (and as we just explained our results apply without any change apart from that of the ancillary variable to those of class VIII). Given the richness of this class we decided to choose a special type of parametrisation of the mappings to analyse, one where a single singularity would control integrability. Our analysis identified all integrable cases and for the non-integrable ones furnished the dynamical degree which was in perfect agreement with a direct calculation thereof. We should point out here that the other known method for the obtention of the dynamical degree, due to Halburd [START_REF] Halburd | Elementary exact calculations of degree growth and entropy for discrete equations[END_REF], cannot be applied in order to identify the integrable cases since the integrability is controlled by a single singularity [START_REF] Ramani | Calculating algebraic entropies: an express method[END_REF]. Moreover, it is important to stress that the ancillary form not only made the calculations involved in the full-deautonomisation method technically feasible, but also that the ancillary form is actually special in the sense that it automatically gives the correct "full-deautonomisation"

]ocnmp[ B. Grammaticos and R. Willox for all the mappings in the class we investigated. No extra terms had to be introduced in order to obtain the dynamical degrees of the mappings we investigated.

Having, once more, affirmed the power of the full-deautonomisation method we can ask ourselves what are the possible next steps in our investigation. One possibility would be to apply the method to higher-order mappings for which little is known to date as far as integrability is concerned. (A first application in this direction can be found in a previous work of the authors in [START_REF] Willox | Full-deautonomisation of a lattice equation[END_REF]). Given the fact that full-deautonomisation seems to work in every case, a different path would be to try to cast the method in some rigorous framework or, at least, explain its uncanny effectiveness. We may address either of these questions in some future work of ours.

Appendix: the QRT canonical forms

In the main body of the article we referred on several occasions to the canonical forms of the QRT mapping and their classification. These are often referred to in various works of the authors but they have never been collected in a single organised and easily consultable list. In order to remedy this problem we decided to give here the full list, which can serve for future reference to those interested in these questions. We shall not present the derivation of the list since the pertaining details can be found in [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equation[END_REF] and we limit ourselves to the presentation of the results. In order to fix the notations we assume that the A 0 matrix for the QRT mapping [START_REF] Quispel | Integrable mappings and soliton equations II[END_REF] has the form

A 0 =   α β γ δ ϵ ζ κ λ µ   , (50)
in the asymmetric case, while in the symmetric case we have δ = β, κ = γ and λ = ζ.

In hindsight it is clear that the numbering of the canonical cases is far from optimal but as it was introduced more than 20 years ago, it can no longer be modified. (In fact the same holds for the choice of the symbols for the elements of the QRT matrices. Following the greek alphabet it would have been logical to have in the third line the letters η, θ and κ but, again, introducing such a change at this late date would be a disruption).

We move now to the classification, which also includes the form of the A 1 matrix for the corresponding mapping.

( 1 ) 8 -

 18 associated discrete Painlevé equation which was first obtained in[START_REF] Ohta | An affine Weyl group approach to the 8-parameter discrete Painlevé equation[END_REF] from purely geometrical arguments.

xn+1 x n = κy 2 n

 2 n+1 + x n = -δy2 n + ϵy n + ζ αy2 n + βy n + γ (51a)y n + y n-1 = -βx 2 n + ϵx n + λ αx 2 n + δx n + κ + λy n + µ αy 2 n + βy n + γ (52a)y n y n-1 = γx 2 n + ζx n + µ αx 2 n + δx n + κ + y n )(y n + x n ) = αy 4 n + (βδ)y 3 n + (γ + κϵ)y 2 n + (λζ)y n + µ αy 2 n + βy n + γ (53a) (x n + y n )(x n + y n-1 ) = αx 4 n -(βδ)x 3 n + (γ + κϵ)x 2 n -(λζ)x n + µ αx 2 n + δx n + κ y n -1)(y n x n -1) = κy 4 n + (δ + λ)y 3 n + (µ + ϵ + α)y 2 n + (β + ζ)y n + γ αy 2 n + βy n + γ (54a) (y n x n -1)(x n y n-1 -1) = γx 4 n + (β + ζ)x 3 n + (µ + ϵ + α)x 2 n + (δ + λ)x n + κ αx 2 n + δx n + κ + y n + 2z)(y n + x n + 2z) (x n+1 + y n )(y n + x n ) = αy 4 n +(4αz+β-δ)y 3 n +(4αz 2 +4βz-2δz-ϵ+γ+κ)y 2 n +(4βz 2 -2ϵz+4γz+λ-ζ)y n +4γz 2 -2ζz+µ αy 4 n +(β-δ)y 3 n +(γ+κ-ϵ)y 2 n +(λ-ζ)y n +µ(55a)(x n + y n + 2z)(x n + y n-1 + 2z) (x n + y n )(x n + y n-1

Table 1 .

 1 Largest roots of the characteristic polynomial (36)

			1		1		1		1
		1	1	1	1	1	1	2.618 2.618 3.732
			1		1		1.722		2.890
	3	1	1	1	1	1.883 1.883 2.966 2.966 3.985
	4		1		1		1.947		2.989
	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .

Acknowledgements

RW would like to acknowledge support from the Japan Society for the Promotion of Science (JSPS), through JSPS grant number 22H01130.

]ocnmp[ B. Grammaticos and R. Willox = αx 4 n +(4αz-β+δ)x 3 n +(4αz 2 -2βz+4δz-ϵ+γ+κ)x 2 n +(4δz 2 -2ϵz+4κz-λ+ζ)x n +4κz 2 -2λz+µ αx 4 n -(β-δ)

The case z = 0 in V (sometimes referred to as V 0 ) is special and leads to the mapping

n + (2βδ)y 2 n + (-ϵ + 2γ)y nζ αy 4 n + (βδ)y 3 n + (γ + κϵ)y 2 n + (λζ)y n + µ (56a)

Case VI This is the only canonical case where the upper left element of the A 1 matrix is not zero. It was introduced so as to correspond to the first form obtained for the discrete analogue of the P VI equation.

A alternate form to VI does exist, one where the upper left matrix element is zero. We usually refer to it as VI ′ .

n +(-δ(z 4 -1) 2 +λz 6 +ζz 4 )x n -κ(z 4 -1) 2 +µz 4 (60b) This completes the list of the cases that allow for a symmetric form. The corresponding equation can be obtained from the equation for x n+1 in the (x, y) system by putting δ = β, κ = γ and λ = ζ, and by replacing x n by x n-1 and y n by x n , while leaving x n+1 unchanged.

The remaining three canonical forms correspond to genuinely asymmetric equations. Case IX (IX)

Case XI (XI)

In [START_REF] Ramani | On the canonical forms of QRT mappings and discrete Painlevé equation[END_REF] it was pointed out that the most general form of the A 1 QRT matrix (once the appropriate homographic transformations have been performed) is

When deautonomised, the corresponding QRT mapping should in principle lead to the most general discrete Painlevé equation, namely the elliptic equation associated to E

8 . However obtaining the latter by direct deautonomisation of this canonical form is a practically impossible task. Thanks to the ancillary representation [START_REF] Ramani | Singularity analysis for difference Painlevé equations associated with the affine Weyl group E8[END_REF] this formidable task becomes almost elementary.