Silvia Voci

Romain Duwald

Stéphane Grass

David J Hayne

Laurent Bouffier

Paul S Francis

Jérôme Lacour

Neso Sojic

Self-enhanced multicolor electrochemiluminescence by competitive electron-transfer processes

come

Introduction

Electrochemiluminescence (ECL) is the light emission resulting from an initial electron-transfer reaction occurring at the electrode surface. [START_REF] Bard | Electrochemical methods[END_REF][START_REF] Bard | Electrogenerated Chemiluminescence[END_REF] The discovery of ECL emission in aqueous solution with an efficient 'coreactant' reagent has led to successfully commercialized bioassays for the detection of clinical biomarkers, biological warfare agents and foodborne pathogens. [START_REF] Liu | [END_REF][4][START_REF] Sojic | Analytical Electrogenerated Chemiluminescence: From Fundamentals to Bioassays[END_REF] The coreactant is a sacrificial molecule that is added to the solution containing the luminophore. In the classic ECL mechanistic pathway, both molecules, the luminophore and the coreactant, have specific and separate functions. The coreactant that is either oxidized or reduced is consumed irreversibly due to a bond-breaking reaction whereas the luminophore is regenerated during the ECL process. 4,[START_REF] Schmittel | [END_REF] The function of the coreactant is to provide sufficiently energetic radicals to undergo exergonic intermolecular electron-transfer with the luminophore in order to reach the electronicallyexcited state. [7][8][9][10] Tri-n-propylamine (TPrA) and several closely related tertiary amines are the most efficient coreactants to enhance ECL intensity in aqueous solution. 9,10 On the other hand, the role of the luminophore is solely to emit in fine light. Thus, electron-transfer is at the core of the ECL phenomenon. A thorough understanding of the underlying principles of the associated inter-and intra-molecular electron-transfer reactions and reactivity is necessary to propose novel ECL strategies.

Since light is the final output of ECL, original concepts and applications have been developed exploiting the different properties of light, such as surface plasmon resonance ECL, 11 circularly-polarized ECL, 12 light-emitting devices, 13,14 photoinduced ECL, 15 and ECL interferometry. 16 Recently, ECL has been used as a microscopy technique to image latent fingerprints, 17 single micro-and nano-particles, [18][19][20][21][22] cells, 23,24 cellular membranes, 25 and membrane proteins. 26,27 Most of these applications rely on the use of ruthenium(II) polypyridyl complexes as luminophores, especially ruthenium(II) tris(bipyridine), and more recently on ortho-metalated or heteroleptic cationic iridium complexes. [28][29][30][31][32][33][34][35] Indeed, the emission of such iridium(III) complexes is tunable at more hypsochromic wavelengths compared to the typical orange/red color resulting from the ruthenium complexes. [31][32][33][34]36 The ability to tune the emission color(s) has generated recently a wide interest due to potential applications ranging from multiplexed bioassays to ECL microscopy. [28][29][30][31][37][38][39][40][41][42] However, the effective implementation of such strategies is constrained by the limited number of luminophores that combine water solubility and a Please do not adjust margins Please do not adjust margins high ECL efficiency in aqueous media. In this context, multicolor ECL emission has been reported using a combination of different organometallic ruthenium(II) and iridium(III) luminophores, mainly in organic media. [28][29][30]41 The concomitant ECL emissions of several luminophores has been described by controlling the energetics of the luminophore. 30,43,44 Ding and coworkers have explored a different approach with a specifically designed iridium complex emitting at three wavelengths in acetonitrile. 45 Nanoparticles have also been used for this purpose. 39 For example, a hybrid system consisting of PbS nanocrystals and a BODIPY capping ligand was proposed to generate highly efficient dual emissions. 46 However, organic molecules offer a promising alternative due to their structural versatility that allows to tune precisely the optical and electronic properties. A large variety of such organic molecules have been designed and prepared to investigate their ECL properties, almost exclusively in organic solvents. [47][48][49][50][51][52][53][54] In this work, we demonstrate that the integration of both functions (i.e. the coreactant, which generates energetic radicals and the luminophore, which is thus "electro-excited" and emits light) can be decoupled and accomplished sequentially in a single molecule in order to control multicolor ECL emissions. Herein, we report the first self-enhanced organic ECL dye that integrates successfully both coreactant and luminophore functions. The molecular scaffold is based on a cationic diaza [4]helicene (Fig. 1a). With the goal of obtaining multicolor ECL emission, we selected the archetypical [Ru(bpy)3] 2+ complex and also two iridium(III) luminophores that emit at shorter wavelengths to avoid Förster resonance energy transfer (Fig. 1b). Hence, we investigate competitive highly exergonic intra-and inter-molecular electron-transfer reactions between different luminophores (Fig. 1). Finally, we demonstrate that concomitant multicolor ECL emission ranging from red to blue with tunable intensity can be obtained by intermolecular electron-transfer processes between the electron-donor helicene radicals and all three electron-acceptor organometallic complexes.

Results and discussion

Cationic helicenes are ortho-fused polyaromatics that combine the main properties of triarylcarbenium ions with the structural features of (chiral) helicenes. 55,56 These dyes and luminophores are readily prepared, 56 and can be further functionalized via versatile late stage synthetic strategies that allow the regioselective introduction of various functional groups at the periphery of the helical core. [57][58][59][60][61] We selected this family of luminophores because it allows fine-tuning not only the photophysical and redox properties of the luminophores but also its solubility in aqueous media. 56,62 More specifically, a diaza [4]helicene bearing a tertiary dimethylamino-terminated group, noted [HEL-AMH] + , was designed and prepared. In this dual molecule, HEL and AMH refer to the [4]diaza core and to the dimethylamino moiety, respectively. It was foreseen that the core helical structure would (i) remain a red-emitter distinct from the emission of model ruthenium and iridium complexes, 57 (ii) integrate easily the coreactant function as a substituent, 58,60 and (iii) exhibit also reasonable solubility in aqueous media. In terms of synthesis, the central helical core was functionalized at position 6 with an ester functional group featuring a dimethylamino moiety as its terminal end. Previously, the synthesis of the diaza [4]helicene carboxylic acid was reported (Scheme 1). 63 Herein, this compound was converted to the corresponding acyl chloride with CO2Cl2 and subjected to a nucleophilic substitution reaction at 0 °C by slow addition of an excess of N,N-dimethylethanol. The corresponding ester [HEL-AMH] + was isolated after purification by flash column chromatography in 74% yield.

Scheme 1 Synthesis of water soluble diaza [4]helicene [HEL-AMH] + .

As expected, the dimethylamino group improved drastically the aqueous solubility of the luminophore (Fig. S9 andS10) and its role as coreactant was tested. The electrochemical Please do not adjust margins Please do not adjust margins characterization of the helicene shows two irreversible oxidation waves in PBS (Fig. 2). The first wave occurs at 0.8 V, which can be attributed to the oxidation of the dimethylamino AMH moiety (reaction 1). Indeed, the irreversible oxidation of TPrA has been reported previously at the same potential in aqueous media and we can assume a mechanism similar to the one described for TPrA. 64 The oxidation of the deprotonated dimethylamino group generates the dication radical [HEL-AMH • ] 2+ (reaction 1). The latter deprotonates rapidly to form locally the monocation radical [HEL-AM • ] + , which is a strong reductant (reaction 2). The second irreversible wave peaking at 1.33 V corresponds to the oxidation of the diaza core. To confirm this assignment, another water soluble diaza [4]helicene [HEL-SO3H] + was also prepared (79% from the carboxylic acid precursor, see Supporting Information). In that case, the sulfonic group is electro-inactive in the investigated potential window and only the oxidation of the diaza core is visible at 1.3 V (Fig. S13). Through a simple comparison of the voltammetric behavior of both [4]diaza complexes, we can therefore assign without ambiguity the two oxidation waves of [HEL-AMH] + : the first and second correspond to the oxidation of the dimethylamino group and of the diaza core (reaction 3), respectively. In acetonitrile, the oxidation of helicene [4]diaza complexes is reversible and it means that the resulting electrogenerated cation is stable in such an organic solvent. In PBS, the oxidation wave of the diaza core is irreversible, indicating that the oxidized radical of the diaza core is not stable over the timescale of the voltammetric experiments. In addition, since the same oxidation potential of the diaza core is obtained with and without the dimethylamino group, it implies that the electronic communication between these groups is negligible. This is further supported by the near identical oxidation potential of TPrA and the dimethyl amino moiety of [HEL-AMH] + . In brief, the two parts of the luminophorecoreactant molecule behave independently from an electrochemical point of view. This decoupling is an essential prerequisite for the ECL mechanism, which involves intermolecular electron-transfer reactions with the organometallic complexes to enable multicolor ECL emission (vide infra). At an applied potential above 1.3 V, both parts of the molecules are oxidized and the resulting trication [HEL • -AMH • ] 3+ intermediate is probably formed (reaction 3) and may deprotonate to form [HEL • -AM • ] 2+ (reaction 4). We can assume that the coreactant mechanism for the amino moiety is analogous to that of the model TPrA reagent (see the detailed structures of the amino radicals in Scheme S1). 64,65 [HEL-AMH] + -e - [HEL-AMH • ] 2+

(1)

[HEL-AMH • ] 2+  [HEL-AM • ] + + H + (2) [HEL-AMH] + -2 e - [HEL • -AMH • ] 3+ (3) [HEL • -AMH • ] 3+  [HEL • -AM • ] 2+ + H +
(4) Then we investigated the ECL properties of this helicene molecule. As shown on Fig. 2, ECL light is only emitted at potentials higher than 1.2 V. When the potential is scanned towards more positive values, ECL emission grew and reached a maximum at 1.32 V. It means that ECL generation requires explicitly the oxidation of the diaza core. ECL is obtained without adding any external coreactant; the dimethylamino moiety acting by itself as a coreactant in the mechanism. In other words, self-enhanced ECL emission is generated by this dual helicene compound. The excited state may be reached by intramolecular (reaction 5) or intermolecular (reaction 6) electron-transfer reactions between the strong radical reductant (denoted -AM • ) and the oxidized diaza core (HEL • ). It then deactivates through the emission of a photon at 650 nm (reactions 7 and 8). In this path, ECL generation requires the simultaneous presence of both radicals to form the excited state. Even if the electrogenerated radicals are not stable during the timescale of the voltammetric experiments, the ECL emission is an indirect proof of their formation at the vicinity of the electrode surface. Both inter-and intra-molecular pathways may be postulated and they depend on the radical lifetimes, their reactivity and concentration, amongst other factors. Indeed, self-enhanced ECL of ruthenium tris-bipyridyl and iridium complexes with different amine reductants has been reported based on intra-and inter-molecular interactions. 45,66 Self-enhanced ECL has been also applied for sensing applications. 42,[67][68][69] In the present case, the resulting ECL spectrum is identical to the fluorescence spectrum (Fig. S14). Therefore, the same excited state is reached by photoexcitation and by self-enhanced ECL. By assuming that the intramolecular coreactant exhibits the same potential as TPrA • after deprotonation, it is possible to calculate the exergonicity of reactions 5 or 6 (ΔG° = E°red(TPrA • ) -E°ox(helicene core) ≈ -3 eV with E°red(TPrA • ) being reported to be -1.7 V vs Ag/AgCl 64,70 ). A direct comparison with the emission wavelength of the [4]helicene core (EFl = 1239.81 / λmax = 1.9 eV) indicates that the excited state can be directly populated.

[HEL • -AM • ] 2+  [HEL*-IM] 2+ (5) 2 [HEL • -AM • ] 2+  [HEL*-AM • ] + + [HEL • -IM] 3+ (6) [HEL*-IM] 2+  [HEL-IM] 2+ + hνred (650 nm) (7) [HEL*-AM • ] +  [HEL-AM • ] + + hνred (650 nm) (8)
where IM represents the iminium product (see the detailed structures in Scheme S1). 64,71 Annihilation ECL of [Ru(bpy)3] 2+ with several iridium(III) complexes has been investigated previously in organic solvents. 28,29 In the present work, our goal was to generate Please do not adjust margins Please do not adjust margins multicolor ECL emission by combining it with self-enhanced ECL.

For this purpose, we added first [Ir(df-ppy)2(pt-R)] + (Fig. 1a, where R is a trioxatridecane chain (TOxT) and terminal squarate amide ethyl ester (Sq)) 72 to an aqueous solution containing already [HEL-AMH] + . The Sq functionality does not change the electrochemical and photophysical properties of the parent luminophore, but provides a convenient attachment point for future works for involving biomolecular labelling and multicolour ECL biodetection. Oxidation of the [Ir(df-ppy)2(pt-R)] + complex occurs at a higher potential than the diaza core and the corresponding ECL emission is obtained at 1.5 V (Fig. S15).

When both reagents are present, blue and red ECL emissions of [Ir(df-ppy)2(pt-R)] + and diaza luminophore are observed concomitantly (Fig. 3). Such a dual ECL emission is achieved without adding any external coreactant. As already mentioned, Förster resonance energy transfer processes are not possible since the emission of [Ir(df-ppy)2(pt-R)] + occurs at lower wavelengths than that of the diaza [4]helicene. It is likewise the same energetic configuration for all the three tested organometallic luminophores. The excited states of both luminophores are reached following the reactions occurring between their respective oxidized forms (i.e. [Ir(df-ppy)2(pt-R)] 2+ and [HEL • -AMH] 2+ ) and the radical -AM • of the modified helicene (reaction 10). For the iridium complex, it corresponds to an intermolecular electron-transfer reaction with the radical -AM • moiety (reaction 10). The coreactant function of the diaza complex is activated and operates via intermolecular mode for [Ir(df-ppy)2(pt-R)] 2+ (reactions 9-10) and/or intramolecular modes for [HEL-AMH] + (reactions 5 and or 6).

[

Ir(df-ppy)2(pt-R)] + -e - [Ir(df-ppy)2(pt-R)] 2+ (9) [Ir(df-ppy)2(pt-R)] 2+ + [HEL • -AM • ] 2+  [Ir(df-ppy)2(pt-R)] + * + [HEL • -IM] 3+ (10) [Ir(df-ppy)2(pt-R)] + *  [Ir(df-ppy)2(pt-R)] + + hνblue (11)
The [HEL-AMH] + molecule, which is a red-emitter, induces also the generation of blue ECL color by [Ir(df-ppy)2(pt-R)] + (reaction 11). Even if the final result is reminiscent of an upconversion process, it is not mechanistically the case here. The blue ECL emission results from a very exergonic electron-transfer reaction from the [HEL • -AM • ] 2+ radical to the oxidized [Ir(df-ppy)2(pt-R)] 2+ . It is not related to an energy-transfer process such as triplet-triplet annihilation. Herein, we solely control and exploit the dual reactivity of the [HEL-AMH] + molecule.

In the previous example, we showed that ECL can be tuned from red to blue by intermolecular electron-transfer processes. Herein, we investigate the ECL of [Ru(bpy)3] 2+ using this original configuration. Indeed, this complex is the model and most widely used ECL luminophore due to its remarkable stability and high ECL efficiency in water. We tested its behavior by increasing its concentration in a solution containing the [HEL-AMH] + compound (Fig. S16). The maximum ECL intensity moved progressively from 650 nm (characteristic of [HEL-AMH] + ) to 620 nm (Fig. 4), corresponding to the emission of [Ru(bpy)3] 2+ . This shift is even more striking in the corresponding normalized ECL spectra (Fig. S17). Moreover, one can observe that the [Ru(bpy)3] 2+ ECL band increases remarkably. A similar ECL mechanism as that described above for the iridium luminophore (reactions 9-10) can be proposed. In a solution containing both luminophores at the same 10 -4 M concentration (green curve, Fig. 4), light intensity is 8-fold higher in comparison to the solution containing only the [HEL-AMH] + (red curve, Fig. 4). When the [Ru(bpy)3] 2+ concentration is increased up to 10 -3 M, ECL intensity is 20.5-fold higher. We checked that addition of [Ru(bpy)3] 2+ does not influence the fluorescence of the [HEL-AMH] + (Fig. S18). Therefore, the enhancement of the [Ru(bpy)3] 2+ ECL band is due to the very efficient intermolecular electron-transfer between both species and to the high ECL efficiency of the [Ru(bpy)3] 2+ luminophore. Indeed, [HEL-AMH] + was found to have an ECL efficiency of 7.3 % in comparison to [Ru(bpy)3] 2+ , which accounts for the dramatic increase in ECL intensity. This creates an intriguing situation in which ECL emission resulting from an intermolecular electron-transfer reaction (i.e. with [Ru(bpy)3] 2+ ) is more efficient than that of the intramolecular process (i.e. [HEL-AMH] + ).

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

Please do not adjust margins

Please do not adjust margins

As shown by the voltammetric study, the two moieties of [HEL-AMH] + are electronically decoupled to a large extent. This enables the two functions to be operated either separately or concomitantly by controlling the applied potential and the energetics of the luminophores. The coreactant function of [HEL-AMH] + is activated at 0.8 V where the dimethylamino group is oxidized. The ECL of this molecule, however, requires oxidation of the diaza core at 1.33 V. Therefore, a wide potential window exists between 0.8 V and 1.33 V where another luminophore can be oxidized and generate ECL before the emission of [HEL-AMH] + . For this purpose, we used the green emitter, [Ir(ppy)2(pt-R)] + (Fig. 1). Oxidation of this iridium complex occurs at 1.14 V (Fig. S19), which in the presence of a suitable coreactant elicits green ECL. By imposing 1.1 V, we observed predominantly the green ECL emitted by the [Ir(ppy)2(pt-R)] + (green curve, Fig. 5). This potential induces the oxidation of [Ir(ppy)2(pt-R)] + as well as of the dimethylamino moiety of [HEL-AMH] + , whereas the diaza core is not oxidized, so the red emission of this luminophore was not observed. The oxidized and deprotonated dimethylamino moiety (-AM • ) acts then only as a coreactant by intermolecular electron-transfer with the oxidized [Ir(ppy)2(pt-R)] 2+ , as described by the sequence of reactions 1, 2, 12, 13 and 14:

[Ir(ppy)2(pt-R)] + -e - [Ir(ppy)2(pt-R)] 2+ (12) [Ir(ppy)2(pt-R)] 2+ + [HEL-AM • ] +  [Ir(ppy)2(pt-R)] + * + [HEL-IM] 2+ (13) [Ir(ppy)2(pt-R)] + *  [Ir(ppy)2(pt-R)] + + hνgreen ( 14
) Under these conditions, only the coreactant function of the helicene molecule is activated and green ECL emission of [Ir(ppy)2(pt-R)] + is solely observed (green curve, Fig. 5). At higher potentials, the red and the green ECL emissions of [HEL-AMH] + and [Ir(ppy)2(pt-R)] + are visible concomitantly. The luminescent function of the helicene molecule is activated as well. The helicene molecule acts both as a coreactant and a luminophore. In this potential region, the diaza core is also oxidized and it reacts with the -AM • moiety radical to give its characteristic red emission. In brief, single or multicolor ECL emission is obtained by controlling the reactivity of the dual [HEL-AMH] + luminophore. It is also noteworthy that the resulting color can also be tuned by adjusting the ratio of luminophores dissolved in solution.

Conclusions

In conclusion, self-enhanced ECL emission is demonstrated in aqueous media by integrating bifunctional ECL features in a single molecular system using a red-emitting diaza [4]helicene functionalized with a dimethylamino moiety. The coreactant and the luminescent functions are decoupled and are activated either separately or simultaneously by controlling the applied potential. Self-enhanced multicolor ECL, from red to blue, via orange and green is obtained by intramolecular and intermolecular electron-transfer reactions between the helicene and a series of organometallic iridium or rutheniumbased complexes. Such competitive paths lead to an intriguing heretofore unreported situation where ECL emission resulting from purely intermolecular electron-transfer reaction with [Ru(bpy)3] 2+ is more efficient than the one from the intramolecular and intermolecular processes. Single or multicolor emissions with tunable intensity are produced by controlling the dual reactivity of the helicene molecule. This multicolor ECL strategy can be extended to other luminophores and coreactants and has important implications for electrontransfer and energy-transfer processes, multiplexed bioassays and imaging applications. Please do not adjust margins Please do not adjust margins

Fig. 1 a

 1 Fig. 1 a) Molecular structures and b) photoluminescence emission spectra of the ECL luminophores. From left to right: [Ir(df-ppy)2(pt-R)] + (blue curve/emitter), [Ir(ppy)2(pt-R)] + (green curve/emitter), [Ru(bpy)3] 2+ (orange curve/emitter) and the dual helicene noted [HEL-AMH] + (red curve/emitter). R = TOxT-Sq (a trioxatridecane chain with squarate amide ethyl ester).

Fig. 2

 2 Fig. 2 Voltammetric (black curve) and ECL (blue curve) signals of 10 -4 M [HEL-AMH] + in PBS on a glassy carbon (GC) electrode. Scan rate: 0.1 V s -1 .

4 |

 4 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

Fig. 3

 3 Fig. 3 ECL spectrum recorded in a PBS solution containing 10 -4 M [HEL-AMH] + and 10 -3 M [Ir(df-ppy)2(pt-R)] + . Applied potential: 1.7 V vs Ag/AgCl/KCl.

Fig. 4

 4 Fig.4 ECL spectra recorded in a PBS solution containing a constant concentration of 10 -4 M [HEL-AMH] + and increasing concentrations of [Ru(bpy)3] 2+ : 0, 1 × 10 -5 M, 3 × 10 -5 M, 6 × 10 -5 M, 1 × 10 -4 M and 1 × 10 -3 M, as indicated by arrow. Applied potential: 1.5 V vs Ag/AgCl/KCl.

Fig. 5

 5 Fig. 5 ECL spectra recorded in a PBS solution containing 10 -4 M [HEL-AMH] + and 10 -3 M [Ir(ppy)2(pt-R)] + when applying either 1.1 V (green curve) or 1.3 V (red curve).

6 |

 6 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx

Acknowledgements

S.V. thanks the Agence Nationale de la Recherche (Neocastip ANR-15-CE09-0015-03) for financial support. P.S.F. and D.J.H. acknowledge the Australian Research Council (DP160103046; DP200102947) for funding. R.D., S.G. and J.L thank the University of Geneva and the Swiss National Science (SNF 200020-184843) for funding and support, and acknowledge the contributions of the Sciences Mass Spectrometry (SMS) platform at the Faculty of Sciences, University of Geneva.

Conflicts of interest

There are no conflicts to declare.


Alin Bostan
email: alin.bostan@inria.fr

Sergey Yurkevich
email: sergey.yurkevich@univie.ac.at

On a Class of Hypergeometric Diagonals

We prove that the diagonal of any finite product of algebraic functions of the form

is a generalized hypergeometric function, and we provide explicit description of its parameters.

The particular case (1 -x -y) R /(1 -x -y -z) corresponds to the main identity of Abdelaziz, Koutschan and Maillard in [1, §3.2]. Our result is useful in both directions: on the one hand it shows that Christol's conjecture holds true for a large class of hypergeometric functions, on the other hand it allows for a very explicit and general viewpoint on the diagonals of algebraic functions of the type above. Finally, in contrast to [1], our proof is completely elementary and does not require any algorithmic help.

1 9 , 4 9 , 5 9 , 1, 1 3 ; t the diagonal of an algebraic power series?

Introduction

Let K be a field of characteristic zero and let g ∈ K[[x]] be a power series in x = (x 1 , . . . , x n ) g(x) = (i1,...,in)∈N n g i1,...,in x i1

1 • • • x in n ∈ K[[x]].
The diagonal Diag(g) of g(x) is the univariate power series given by Diag(g) := j≥0 g j,...,j t j ∈ K[[t]].

A power series h(x) in K [[x]] is called algebraic if there exists a non-zero polynomial P (x, T ) ∈ K[x, T ] such that P (x, h(x)) = 0; otherwise, it is called transcendental.

If g(x) is algebraic, then its diagonal Diag(g) is usually transcendental; however, by a classical result by Lipshitz [START_REF] Lipshitz | The diagonal of a D-finite power series is D-finite[END_REF], Diag(g) is D-finite, i.e., it satisfies a non-trivial linear differential equation with polynomial coefficients in K[t]. Equivalently, the coefficients sequence (g j,...,j ) j≥0 of Diag(g) is P-recursive, i.e., it satisfies a linear recurrence with polynomial coefficients (in the index j).

When a P-recursive sequence satisfies a recurrence of order 1, we say that it is hypergeometric. An important class of power series, whose coefficients sequence is hypergeometric by design, is that of generalized hypergeometric functions. Let p, q ∈ N and a 1 , . . . , a p and b 1 , . . . , b q be rational 1 arXiv:2008.12809v1 [math.NT] 28 Aug 2020 numbers such that b i + j = 0 for any i, j ∈ N. The generalized hypergeometric function p F q with parameters a 1 , . . . , a p and b 1 , . . . , b q is the univariate power series in K[[t]] defined by p F q ([a 1 , . . . , a p ], [b 1 , . . . , b q ]; t) := j≥0

(a 1 ) j • • • (a p ) j (b 1 ) j • • • (b q ) j t j j! ,
where (x) j := x(x + 1) • • • (x + j -1) is the rising factorial.

We are interested in this article by the following (dual) questions:

(i) what are the algebraic power series g(x) whose diagonal Diag(g) is a generalized hypergeometric function p F q ?1

(ii) what are the hypergeometric sequences (a j ) j≥0 whose generating functions j≥0 a j t j can be written as diagonals of algebraic power series?

Already for n ∈ {1, 2} these questions2 are non-trivial. The classes of diagonals of bivariate rational power series and of algebraic power series coincide [START_REF] Pólya | Sur les séries entières, dont la somme est une fonction algébrique[END_REF][START_REF] Furstenberg | Algebraic functions over finite fields[END_REF]. Hence, questions (i) and (ii) contain as a sub-question the characterization of algebraic hypergeometric functions. This problem was only recently solved in a famous paper by Beukers and Heckman [START_REF] Beukers | Monodromy for the hypergeometric function n F n-1[END_REF].

Another motivation for studying questions (i) and (ii) comes from the following conjecture, formulated in [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF][START_REF] Christol | Globally bounded solutions of differential equations[END_REF]:

Christol's conjecture. If a power series f ∈ Q[[t]
] is D-finite and globally bounded (i.e., it has non-zero radius of convergence in C and

β •f (α•x) ∈ Z[[x]] for some α, β ∈ Z), then f = Diag(g) for some n ∈ N and some algebraic power series g ∈ Q[[x 1 , . . . , x n ]].
Christol's conjecture is still largely open, even in the particular case when f is a generalized hypergeometric function. In this case, it has been proved [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF][START_REF] Christol | Globally bounded solutions of differential equations[END_REF] in two extreme subcases: when all the bottom parameters b i are integers (case of "minimal monodromy weight", in the terminology of [START_REF] Christol | Diagonals of rational fractions[END_REF]) and when they are all non-integers (case of "maximal monodromy weight"). In the first extremal case, the proof is based on the observation that

p F q ([a 1 , . . . , a p ], [1, . . . , 1]; t) = (1 -t) -a1 • • • (1 -t) -ap , ( 1
)
where denotes the Hadamard (term-wise) product, and on the fact that diagonals are closed under Hadamard product [START_REF] Christol | Diagonales de fractions rationnelles[END_REF]Prop. 2.6]. In the second extremal case, it is based on the equivalence between being globally bounded and algebraic; this equivalence, proved by Christol [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF][START_REF] Christol | Globally bounded solutions of differential equations[END_REF], is itself based on [START_REF] Beukers | Monodromy for the hypergeometric function n F n-1[END_REF]. The other cases (of "intermediate monodromy weight") are widely open. A first explicit example of this kind, itself still open as of today, was given by Christol himself as soon as 1987 [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF]§VII]:

Is f (t) = 3 F 2
Two decades later, Bostan et al. [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity[END_REF][START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF] produced a large list of about 200 similar 3 F 2 (globally bounded) functions, which are potential counter-examples to Christol's conjecture (in the sense that, like 3 F 2 ([1/9, 4/9, 5/9], [1, 1/3]; t), they are not easily reducible to the two known extreme cases, via closure properties of diagonals, e.g., with respect to Hadamard products). This year, Abdelaziz, Koutschan and Maillard [1, §3] managed to show that two members of that list, namely

t = Diag (1 -x -y) 1/3 1 -x -y -z (2)
and

3 F 2 1 9 , 4 9 , 7 9 , 1, 1 3 ; 27 t = Diag (1 -x -y) 2/3 1 -x -y -z . ( 3
)
They moreover gave in [1, §3.2] the following extension of identities ( 2) and (3), to any R ∈ Q:

3 F 2 1 -R 3 , 2 -R 3 , 3 -R 3 , [1, 1 -R]; 27 t = Diag (1 -x -y) R 1 -x -y -z . ( 4
)
A common feature of the identities (2) and (3) (and their generalization (4)) is that the top parameters are in arithmetic progression, as opposed to Christol's initial example. However, they are the first known examples of generalized hypergeometric functions with intermediate monodromy weight, not reducible to the two known extreme cases, and which are provably diagonals.

Our first result extends identity (4) to a much larger class of (transcendental) generalized hypergeometric functions.

Theorem 1. Let R, S ∈ Q and n, N ∈ N such that S = 0 and 0 ≤ n ≤ N . Set s := N -n and Q := S -R. Then the generalized hypergeometric function

N +s F N +s-1 Q N , Q + 1 N , . . . , Q + N -1 N , S s , . . . , S + s -1 s , Q s , . . . , Q + s -1 s , 1, . . . , 1 ; N N t is equal to the diagonal Diag (1 -x 1 -• • • -x n ) R (1 -x 1 -• • • -x N ) S .
Note that identity (4) corresponds to the particular case (n, N, S) = (2, 3, 1) of Theorem 1. The proof of (4) given in [1, §3.2] relies on an algorithmic technique called creative telescoping [START_REF] Koutschan | Creative telescoping for holonomic functions[END_REF], which works in principle3 on any diagonal of algebraic function, as long as the number max(n, N ) of indeterminates is fixed. Our identity in Theorem 1 contains a number of indeterminates which is itself variable, hence it cannot be proved by creative telescoping in this generality. In §2 we offer instead a direct and elementary proof.

In §3 we will further generalize Theorem 1 in two distinct directions. The first extension (Theorem 3) shows that the diagonal of the product of an arbitrary number of arbitrary powers of linear forms of the type 1 -x 1 -• • • -x m is again a generalized hypergeometric function. The second extension (Theorem 4) shows that under a condition on the exponents the same stays true if the product is multiplied with another factor of the form (1

-x 1 -• • • -x m-2 -2 x m-1 ) b .
For instance, when restricted to m = 3 variables, these results specialize respectively into the following identities: Theorem 2. For any R, S, T ∈ Q, we have:

Diag (1 -x) R (1 -x -y) S (1 -x -y -z) T = (5) 6 F 5 -(R + S + T ) 3 , 1 -(R + S + T ) 3 , 2 -(R + S + T ) 3 , -(S + T ) 2 , 1 -(S + T ) 2 , -T ; -(R + S + T ) 2 , 1 -(R + S + T ) 2 , -(S + T ), 1, 1 ; 27t and
Diag (1 -x) R (1 -x -2 y) S (1 -x -y -z) -1 = (6) 4 F 3 1 -(R + S) 3 , 2 -(R + S) 3 , 3 -(R + S) 3 , 1 -S 2 ; 1 -(R + S) 2 , 2 -(R + S) 2 , 1 ; 27t .
Note that (6) generalizes and explains the two following identities observed in [1, Eq. ( 30)-( 31)]

3 F 2 1 9 , 4 9 , 7 9 , 1, 2 3
;

27 t = Diag (1 -x -2 y) 2/3 1 -x -y -z (7)
and

3 F 2 2 9 , 5 9 , 8 9 , 1, 5 6
;

27 t = Diag (1 -x -2 y) 1/3 1 -x -y -z . ( 8
)
Once again, our proofs of the (generalizations of) identities ( 5) and ( 6) are elementary, and do not rely on algorithmic tools. One may wonder if other generalizations are possible, for instance if the coefficient 2 can be replaced by a different one in [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF]. The following example shows that this is not the case. Let

U (t) = Diag 3 √ 1 -ax 1 -x -y .
Then, the coefficients sequence (u j ) j≥0 of U (t) satisfies the second-order recurrence relation

2 a 2 (6 n + 5) (3 n + 1) u n + 9 (a -1) (n + 2) (n + 1) u n+2 = 3 (n + 1) 3 a 2 + 4 a -4 n + 2 a 2 + 18 a -18 u n+1 .
When a ∈ {0, 1, 2}, the sequence (u j ) j≥0 also satisfies a shorter recurrence, of order 1, as shown by our main results. In these cases, U (t) is a hypergeometric diagonal. When a / ∈ {0, 1, 2}, the secondorder recurrence is the minimal-order satisfied by (u j ) j≥0 , hence U (t) is not a hypergeometric diagonal. This can be proved either using the explicit identity

U (t) = 3 a/2 1 -4 t + 1 -a/2 (1 -4 t) 3 2
, or by using the general approach in [9, §5]. An apparent weakness of our results is that they only provide examples with parameters in (unions of) arithmetic progressions. This is true, as long as identities are used alone. But symmetries may be broken by combining different identities and using for instance Hadamard products. As an illustration, by taking the Hadamard product in both sides of the following identities

3 F 2 Q 3 , Q + 1 3 , Q + 2 3 , [1, Q] ; t = Diag (1 -x1 3 -x2 3 ) 1-Q 1 -x1 3 -x2 3 -x3 3 and 2 F 1 Q 6 , Q + 3 6 , Q 3 ; t = Diag (1 -x4 2 -x5 2 ) 1-Q/3 1 -x4 2 -x5 2 ,
both particular cases of Theorem 1, one deduces that the non-symmetric hypergeometric function

4 F 3 Q 6 , Q 6 + 1 2 , Q 3 + 1 3 , Q 3 + 2 3 ; [1, 1, Q] ; t is equal to the following diagonal Diag 1 - x 1 3 - x 2 3 1-Q 1 - x 4 2 1-Q 3 1 - x 1 3 - x 2 3 - x 3 3 -1 1 - x 4 2 - x 5 2 -1
.

Similarly, the non-symmetric hypergeometric function

4 3 F 2 Q 6 , Q 6 + 1 2 , Q 3 + 1 2 ; 1, 2 Q 3 ; t is equal to the following diagonal Diag 1 - x 1 2 1-Q 3 1 - x 3 2 1-2 Q 3 1 - x 1 2 - x 2 2 -1 1 - x 3 2 - x 4 2 -1
.

A natural question is to prove (or, disprove) whether Christol's 3 F 2 can be obtained in such a way.

As a final remark, one should not think that every generalized hypergeometric function which is a diagonal needs to have a representation like in our Theorems 1 or 2. For instance, the diagonal [START_REF] Bostan | Diagonals of rational functions and selected differential Galois groups[END_REF] Diag

1 1 -(1 + w)(x + y + z) is equal to the generalized hypergeometric function 4 F 3 1 3 , 1 3 , 2 3 , 2 3 ; 1, 1, 1 2 ; 729 4 t = 1 + 18 t + 1350 t 2 + • • • ,
which is seemingly not of the form covered by any of our results.

Proof of Theorem 1

Proof. By definition, for k ∈ N, the coefficient of t k in the hypergeometric function is given by

(Q/N ) k ((Q + 1)/N ) k • • • ((Q + N -1)/N ) k (S/s) k • • • ((S + s -1)/s) k (Q/s) k • • • ((Q + s -1)/s) k (k!) N N N k .
Now we extract the coefficient of t k from the diagonal. First note that

Diag (1 -x 1 -• • • -x n ) R (1 -x 1 -• • • -x N ) S (t) = Diag (1 + x 1 + • • • + x n ) R (1 + x 1 + • • • + x N ) -S ((-1) N t).
Recall the definition of the multinomial coefficient: for m ∈ N and α 1 , . . . , α n ∈ N we set

m α 1 , . . . , α n := m! α 1 ! • • • α n ! , if |α| := α 1 + • • • + α n = m.
Otherwise the multinomial coefficient is defined to be zero. By the multinomial theorem we have:

g(x) := (1 + x 1 + • • • + x n ) R (1 + x 1 + • • • + x N ) -S = i≥0, α∈N n R i i α 1 , . . . , α n x α1 1 • • • x αn n j≥0, β∈N N -S j j β 1 , . . . , β N x β1 1 • • • x β N N = i,j≥0, α∈N n ,β∈N N R i -S j i α 1 , . . . , α n j β 1 , . . . , β N x α1+β1 1 • • • x αn+βn n x βn+1 n+1 • • • x β N N . Now for k ∈ N we can extract the coefficient, denoted [x k ]g(x), of x k 1 • • • x k N in g(x): [x k ](1 + x 1 + • • • + x n ) R (1 + x 1 + • • • + x N ) -S = α,γ∈N n αi+γi=k R |α| -S |γ| + (N -n)k |α| α 1 , . . . , α n |γ| + (N -n)k γ 1 , . . . , γ n , k, . . . , k = α∈N n R |α| -S N k -|α| |α| α 1 , . . . , α n N k -|α| k -α 1 , . . . , k -α n , k, . . . , k = α∈N n R |α| -S N k -|α| |α| α 1 , . . . , α n nk -|α| k -α 1 , . . . , k -α n N k -|α| k, . . . , k, nk -|α| .
Recall the multinomial Chu-Vandermonde identity [28, Eq. (1.4)]:

m 1 + m 2 γ 1 , . . . , γ n = α∈N n m 1 α 1 , . . . , α n m 2 γ 1 -α 1 , . . . , γ n -α n .
Using it twice we have

α∈N n R |α| -S N k -|α| |α| α 1 , . . . , α n nk -|α| k -α 1 , . . . , k -α n N k -|α| k, . . . , k, nk -|α| = j≥0 R j -S N k -j N k -j k, . . . , k, nk -j |α|=j j α 1 , . . . , α n nk -j k -α 1 , . . . , k -α n = j≥0 R j -S N k -j N k -j k, . . . , k, nk -j nk k, . . . , k = (N -n)k k, . . . , k nk k, . . . , k j≥0 R j -S N k -j N k -j nk -j = (N -n)k k, . . . , k nk k, . . . , k -S (N -n)k j≥0 R j -S -(N -n)k nk -j = (N -n)k k, . . . , k nk k, . . . , k -S (N -n)k R -S -(N -n)k nk .
Henceforth, in order to prove Theorem 1, it suffices to ensure that

(-1) N k (N -n)k k, . . . , k nk k, . . . , k -S (N -n)k R -S -(N -n)k nk ! = (Q/N ) k ((Q + 1)/N ) k • • • ((Q + N -1)/N ) k (S/s) k • • • ((S + s -1)/s) k (Q/s) k • • • ((Q + s -1)/s) k (k!) N N N k .
However this is easily verified by canceling obvious factors and then using the fact that

(a/b) k ((a + 1)/b) k • • • ((a + b -1)/b) k • b bk = (a) bk .

General case

This section contains several parts: first we introduce in §3.1 and §3.2 some notation and state the two general Theorems 3 and 4. Then we explain them in §3.3 by means of four examples, showing that both Theorem 1 and Theorem 2 are special cases. Further, we continue in §3.4 with several lemmas and their proofs. Finally, the general theorems are proven in §3.5 and §3.6.

First Statement

Let N ∈ N \ {0} and b 1 , . . . , b N ∈ Q with b N = 0. We want to prove that the diagonal of

R(x 1 , . . . , x N ) := (1 + x 1 ) b1 (1 + x 1 + x 2 ) b2 • • • (1 + x 1 + • • • + x N ) b N (9)
can be expressed as a hypergeometric function. For each k = 1, . . . , N we define the tuple

u k := B(k) N -k + 1 , B(k) + 1 N -k + 1 , . . . , B(k) + N -k N -k + 1 , where B(k) := -(b k + • • • + b N ). For k = 1, . . . , N -1 we set v k := B(k) N -k , B(k) + 1 N -k , . . . , B(k) + N -k -1 N -k .
Moreover set v N := (1, 1, . . . , 1) with exactly N -1 ones. It follows by construction that the lengths of the tuples u := (u 1 , . . . , u N ) and

v := (v 1 , . . . , v N )
are given by M := N + • • • + 2 + 1 = N (N + 1)/2 and M -1 respectively. We have the following generalization of Theorem 1:

Theorem 3. It holds that Diag(R(x 1 , . . . , x N )) = M F M -1 (u; v; (-N ) N t).

Second Statement

Let

N ∈ N \ {0} and b 1 , . . . , b N ∈ Q with b N = 0. Assume that b N = 0 and b N -1 + b N = -1.
We will prove that, for any b ∈ Q, we can express

(1 + x 1 + • • • + x N -2 + 2 x N -1 ) b • R(x 1 , . . . , x N )
as a hypergeometric function as well. Again, let B(k

) := -(b k + • • • + b N ).
For each k = 1, . . . , N -2 we define the tuple

u k := B(k) -b N -k + 1 , B(k) -b + 1 N -k + 1 , . . . , B(k) -b + N -k N -k +
v k := B(k) -b N -k , B(k) -b + 1 N -k , . . . , B(k) -b + N -k -1 N -k ,
and v N -1 := (1, 1, . . . , 1) with exactly N -1 ones. It follows by construction that the lengths of the tuples u := (u 1 , . . . , u N ) and

v := (v 1 , . . . , v N -1 )
are given by M -

1 = N + • • • + 4 + 3 + 1 + 1 = N (N + 1)/2 -1 and M -2 respectively. Theorem 4. It holds that Diag((1 + x 1 + • • • + x N -2 + 2 x N -1 ) b • R(x 1 , . . . , x N )) = M -1 F M -2 (u; v; (-N ) N t).

Examples

Let us list some examples of the general theorems and draw the connection to previous statements.

1. First we emphasize that Theorem 1 follows promptly from the more general Theorem 3 by letting all b j = 0 except b n = R and b N = -S. Clearly, the change x → -x in the algebraic function is reflected by the change t → (-1) N t in its diagonal.

2. Letting N = 3 in Theorem 3 we obtain immediately the first part of Theorem 2

Diag (1 + x) R (1 + x + y) S (1 + x + y + z) T = 6 F 5 -(R + S + T ) 3 , -(R + S + T ) + 1 3 , -(R + S + T ) + 2 3 , -(S + T ) 2 , -(S + T ) + 1 2 , -T ; -(R + S + T ) 2 , -(R + S + T ) + 1 2 , -(S + T ), 1, 1 ; -27t .
3. If moreover T = -1 in the previous example, we achieve a cancellation of the last parameter and are left with

Diag (1 + x) R (1 + x + y) S 1 + x + y + z = 5 F 4 1 -(R + S) 3 , 2 -(R + S) 3 , 3 -(R + S) 3 , 1 -S 2 , 2 -S 2 ; 1 -(R + S) 2 , 2 -(R + S) 2 , 1 -S, 1 ; -27t .
Diag (1 + x + 2 y) T • (1 + x) R (1 + x + y + z) -1 = 4 F 3 1 -(R + T ) 3 , 2 -(R + T ) 3 , 3 -(R + T ) 3 , 1 -T 2 ; 1 -(R + T ) 2 , 2 -(R + T ) 2 , 1 ; -27t .

Lemmas and Proofs

In this section we will state and prove necessary lemmas for the proofs of Theorems 3 and 4.

Lemma 1. Let N be a positive integer and b

1 , . . . , b N ∈ Q such that b N = 0. It holds that [x k1 1 • • • x k N N ](1 + x 1 ) b1 (1 + x 1 + x 2 ) b2 • • • (1 + x 1 + • • • + x N ) b N = b N k N b N -1 + b N -k N k N -1 • • • b 1 + • • • + b N -k N • • • -k 2 k 1 .
This result contains the core identity of the present paper, since it enables the connection between the algebraic functions R(x) of the form (9) and hypergeometric sequences. It can be proven in two ways: a direct approach generalizes the proof of Theorem 1 and works by multiplying the left-hand side out using the multinomial theorem, picking the needed coefficient and reducing the sum using the Chu-Vandermonde identity several times. This procedure is rather tedious and not instructive, therefore we present a combinatorially inspired proof.

Proof. We have

[x k1 1 • • • x k N N ](1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N ) b N = [x k1 1 • • • x k N N ](1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N -1 ) b N -1 • x b N N 1 + 1 + x 1 + • • • + x N x N b N = [x k1 1 • • • x k N N ](1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N -1 ) b N -1 • x b N N j≥0 b N j 1 + x 1 + • • • + x N -1 x N j . Because (1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N -1 ) b N -1 does not depend on x N , we must have k N = b N -j.
This reduces the sum to one term, namely j = b N -k N , and we obtain

[x k1 1 • • • x k N N ](1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N ) b N = b N k N • [x k1 1 • • • x k N -1 N -1 ](1 + x 1 ) b1 • • • (1 + x + • • • + x N -1 ) b N -1 (1 + x 1 + • • • + x N -1 ) b N -k N = b N k N • [x k1 1 • • • x k N -1 N -1 ](1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N -2 ) b N -2 (1 + x 1 + • • • + x N -1 ) b N -1 +b N -k N .
Now the claim follows by iteration.

Note that Lemma 1 shares some similarities with Straub's [25, Theorem 3.1], which provides explicit expressions of rational power series of the form

(1 + x 1 + • • • + x λ1 )(1 + x λ1+1 + • • • + x λ1+λ2 ) • • • (1 + x λ1+•••+λ -1 + • • • + x N ) -α • x 1 x 2 • • • x N -1 .
In Lemma 1, we allow products of linear forms with arbitrary exponents but no term α •x 1 x 2 • • • x N , while in [25, Theorem 3.1] the linear forms have disjoint variables and appear at exponent 1. Setting α = 0 in Straub's formula also yields a product of binomial coefficients.

It is legitimate to wonder whether there is a common generalization of Lemma 1 and Thm. 3.1 in [START_REF] Straub | Multivariate Apéry numbers and of rational functions[END_REF]. For instance, one may ask for which values of α is the diagonal

Diag √ 1 -x (1 -y) -αxy -1 = 1 + (α + 1/2) t + α 2 + 2 α + 3/8 t 2 + • • •
hypergeometric? For a general α, the minimal recurrence satisfied by the coefficients of the diagonal is of order 4, for α = ±i/2 it is of order 3, and it seems that the only rational value of α for which there exists a shorter recurrence is α = 0, in which case the diagonal is hypergeometric. Now we want to verify a similar statement for the situation as in Theorem 4, so the case where we deal with the coefficient sequence of

(1 + x 1 + • • • + x N -2 + 2 x N -1 ) b • R(x 1 , . . . , x N ).
We lay the grounds for a lemma similar to Lemma 1, by starting with a rather surprising identity.

Lemma 2. Let k ∈ N and b ∈ Q arbitrary. It holds that [x k ] (1 + 2x) b (1 + x) k+1 = 4 k (b -1)/2 k
Proof. First notice that for arbitrary a, b we can compute

[x k ](1 + 2x) b (1 + x) a = [x k ]   i≥0 2 i b i x i     j≥0 a j x j   = k j=0 2 j b j a k -j .
So we set a = -(k + 1) and obtain

[x k ] (1 + 2x) b (1 + x) k+1 = k j=0 2 j b j -k -1 k -j = 2 k k j=0 (-1) j 2 -j b k -j k + j k .
It remains to prove the following identity

5 k j=0 (-2) -j b k -j k + j k = 2 k (b -1)/2 k . ( 10
)
To do this, we note that

k j=0 (-1) j b k -j k + j k u j = b k 2 F 1 ([-k, k + 1]; [b + 1 -k]; u),
and

2 F 1 ([-k, k + 1]; [b + 1 -k]; 1/2) = Γ((b + 1 -k)/2)Γ((b + 2 -k)/2) Γ((b + 1 -2k)/2)Γ((b + 2)/2) = 2 k (b-1)/2 k b k
, by Kummer's identity [START_REF] Kummer | Über die hypergeometrische Reihe 1 + α•β 1•γ x + α(α+1)β(β+1) γ(γ+1) . . . . (Fortsetzung)[END_REF]Eq. 3,p. 134].

The proof above explains the special role of the coefficient a = 2 mentioned in the introduction: it is one of the few values, along with 1 and -1, for which there exists a closed form expression for the evaluation of a 2 F 1 hypergeometric function of u at u = 1/a. Now we can step forward and prove the essential lemma for Theorem 4. Note that contrary to Lemma 1 the following statement is purely about diagonal coefficients and not for general exponents anymore. Except for the missing factor

∈ Q such that b N -1 + b N = -1. For any b ∈ Q the coefficient of x k 1 • • • x k N in (1 + x 1 + • • • + x N -2 + 2 x N -1 ) b • (1 + x 1 ) b1 • • • (1 + x 1 + • • • + x N ) b N
is given by

4 k (b -1)/2 k b N k • b N -2 + b N -1 + b N + b -2k k • • • b 1 + • • • + b N + b -(N -1)k k .
for all k = 1, . . . , N -1. Clearly

V N := N -1 i=1 (v N i ) n = (n!) N -1 . We deduce that [t n ] M F M -1 (u; v; (-N ) N t) = (-N ) nN n! N i=1 U i V i = (-1) nN (n!) N N i=1 (-b i -• • • -b N ) (N -i+1)n (-b i -• • • -b N ) (N -i)n = (-1) nN (n!) N N i=1 (-b i -• • • -b N + (N -i)n) n .
The claim of Theorem 3 follows from the fact that

(-1) n (-b k -• • -b N + (N -k)n) n n! = (-1) n -b k -• • • -b N + (N -k + 1)n -1 n = b k + • • • + b N -(N -k)n n .

Proof of Theorem 4

The proof of Theorem 4 is very similar: we will use Lemma 3 and the same reasoning as before.

The only difference lies in the fact that because the hypergeometric function has one parameter less, we need to redefine U N -1 , V N -1 and V N . Recall that the denominator of U k was given by (N -k + 1) (N -k+1)n and it cancelled with the denominator of V k-1 . In the present case U N -1 will have no denominator and therefore 2 2n from V N -2 survives. This fits with the 4 k in the statement of Lemma 3 and is another indicator for the importance and essence of the constant a = 2.

Using Lemma 3 we obtain the coefficient of t n for any n ∈ N on the left-hand side:

[t n ]Diag((1 + x 1 + • • • + 2 x N -1 + x N ) b R(x 1 , . . . , x N )) = 4 n (b -1)/2 n b N n • b N -2 + b N -1 + b N + b -2n n • • • b 1 + • • • + b N + b -(N -1)n n .
By the same reasoning as before, we have for all k = 1, . . . , N -2, N

U k := N -k+1 i=1 (u k i ) n = (-b k -• • • -b N -b) (N -k+1)n (N -k + 1) (N -k+1)n ,
and similarly

V k := N -k i=1 (v k i ) n = (-b k -• • • -b N -b) (N -k)n (N -k) (N -k)n , for k = 1, . . . , N -2. Clearly V N -1 := N -1 i=1 (v N -1 i ) n = (n!) N -1 and we set V N := 1. Moreover, this time we have U N -1 := (u N -1 ) n = ((1 -b)/2) n . Figure 1: A pictorial proof of the algebraicity of 3 F 2 1-R 3 , 2-R 3 , 3-R 3 , [ 1 2 , 1 -R]; t for R ∈ { 1 3 , 2 3
}. There are ϕ(18) = 6 conditions to check, which lead to two distinct interlacing configurations.

Henceforth in order to introduce new integer parameters on the top, while not creating equally many on the bottom, it is only possible to choose -b N integer. Therefore in order to archive monodromy weight zero -a necessary condition for algebraicity of H(t) -we need to have N -1 ≤ 1.

From the same argument it follows that in the case N -1 = 1, we need to have -b N ∈ Z.

Obviously for N = 1 the diagonal is algebraic, so it remains to prove that, conversely, when -b 2 =: S is an integer and b 1 =: R ∈ Q arbitrary, then the diagonal in

Diag (1 -x 1 ) R (1 -x 1 -x 2 ) S = 3 F 2 S -R 2 , S -R + 1 2 , S , [1, S -R] ; 4 t . ( 11
)
is an algebraic function. If R is an integer too, this follows from by [START_REF] Pólya | Sur les séries entières, dont la somme est une fonction algébrique[END_REF][START_REF] Furstenberg | Algebraic functions over finite fields[END_REF]. In the general case, one can rewrite the 3 F 2 in (11) as the Hadamard product

3 F 2 S -R 2 , S -R + 1 2 , S , [1, S -R] ; t = 2 F 1 S -R 2 , S -R + 1 2 , [S -R] ; t (1 -t) -S . ( 12
)
The 2 F 1 is algebraic as it corresponds to Case I in Schwarz's table [START_REF] Schwarz | Über diejenigen Fälle, in welchen die Gaußische hypergeometrische Reihe einer algebraische Funktion ihres vierten Elementes darstellt[END_REF]. Since S is an integer, (1-t) -S is a rational function. We conclude by applying Jungen's theorem [START_REF] Jungen | Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence[END_REF]Thm. 8]: the Hadamard product of an algebraic and a rational function is algebraic, see also [START_REF] Stanley | Enumerative combinatorics[END_REF]Prop. 6.1.11].

Hadamard grade

Recall that the Hadamard grade [START_REF] Allouche | Hadamard grade of power series[END_REF] of a power series S(t) is the least positive integer h = h(S) such that S(t) can be written as the Hadamard product of h algebraic power series, or ∞ if no such writing exists. Since algebraic power series are diagonals [14, §3]

• • • (1 + x 1 + • • • + x N ) b N
), or at least decide if it is finite or not. For instance, by Theorem 1, when S = 1, R = 1/2, the diagonal

Diag (1 -x 1 -x 2 ) R (1 -x 1 -x 2 -x 3 ) S
has Hadamard grade 2 (hence is a diagonal) for (A, B, C; D) in the following set (1/4, 3/8, 7/8; 1/3), 1/4, 3/8, 7/8; 2/3), (1/8, 5/8, 3/4; 1/3), (1/8, 5/8, 3/4; 2/3), (1/9, 4/9, 7/9; 1/2), (1/9, 4/9, 7/9; 1/3), (1/9, 4/9, 7/9; 1/4), (1/9, 4/9, 7/9; 1/6), (1/9, 4/9, 7/9; 2/3), (1/9, 4/9, 7/9; 3/4), (2/9, 5/9, 8/9; 1/2), (2/9, 5/9, 8/9; 1/3), (2/9, 5/9, 8/9; 1/4), (2/9, 5/9, 8/9; 2/3), (2/9, 5/9, 8/9; 3/4), (2/9, 5/9, 8/9; 5/6) .

Note that the authors of [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity[END_REF][START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF] produced in 2011 a list of 116 potential counter-examples to Christol's conjecture; they displayed a sublist of 18 cases in the preprint [5, Appendix F], of which they selected 3 cases that were published in [6, §5.2]. As of today, to our knowledge, the 3 cases in [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions[END_REF] are still unsolved 7 , while 2 of the 18 cases in [START_REF] Bostan | Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity[END_REF] have been solved in [START_REF] Abdelaziz | On Christol's conjecture[END_REF] (in red, above) and 2 others in the current paper (in orange, above). From the list of 116 cases, only 2 were previously solved, in [START_REF] Abdelaziz | On Christol's conjecture[END_REF]. Corollary 2 solves 14 cases more, raising the number of solved cases to 16 (out of 116).

1 and set u N - 1 :

 11 = -(b N -1 + b N + b)/2 = (1 -b)/2 and u N := -b N . Moreover, for k = 1, . . . , N -2 we set

4 .

 4 Comparing with the similar situation of Theorem 4 in the case N = 3 and b N -1 = -1-b N = 0, we see that a family of 4 F 3 functions remains and covers the second statement of Theorem 2:

Lemma 3 .

 3 b N -1 +b N -k k and the new two factors 4 k and (b-1)/2 k the statement is completely analogous. Let N be a positive integer and b 1 , . . . , b N

  , and diagonals are closed under Hadamard product [11, Prop. 2.6], any power series with finite Hadamard grade is a diagonal [2, Thm. 7]. Conversely, it is not clear whether diagonals always have finite Hadamard grade 6 . A natural question in relation with Corollary 1 is the following: given b 1 , . . . , b N ∈ Q, determine the Hadamard grade of Diag((1 + x 1 ) b1

Note that a necessary condition is that q = p -1, since the radius of convergence must be finite and non-zero.

From an algorithmic viewpoint, questions (i) and (ii) are very different in nature: while (i) is decidable (given an algebraic power series, one can decide if its diagonal is hypergeometric, for instance by combining the algorithms in[START_REF] Bostan | Creative telescoping for rational functions using the Griffiths-Dwork method[END_REF] and[START_REF] Petkovšek | Hypergeometric solutions of linear recurrences with polynomial coefficients[END_REF]), the status of question (ii) is not known (does there exist an algorithm which takes as input a hypergeometric sequence and outputs an algebraic series whose diagonal is the generating function of the input sequence?).

Creative telescoping algorithms, such as the one in[START_REF] Bostan | Creative telescoping for rational functions using the Griffiths-Dwork method[END_REF], compute a linear differential equation for Diag(g(x)). This equation is converted on a linear recurrence, whose hypergeometric solutions can be computed using Petkovšek's algorithm[START_REF] Petkovšek | Hypergeometric solutions of linear recurrences with polynomial coefficients[END_REF]. Note that the complexity (in time and space) of these algorithms increase with n, N, R and S.

Amusingly, the above 3 F 2 is not only asymmetric, but it also shares another similarity with Christol's example: the sum of two of the three top parameters is equal to the third one. This patters occurs in several other examples.

Note that identity[START_REF] Christol | Fonctions hypergéométriques bornées[END_REF] could alternatively be proven by using Zeilberger's creative telescoping algorithm[START_REF] Zeilberger | The method of creative telescoping[END_REF], or derived from identity(3.42) in[15, p. 27] by setting 2n -x = b, multiplying with 2 k and reverting the summation.

There exist diagonals of any prescribed finite grade [22, Cor. 1 and 2], assuming the Rohrlich-Lang conjecture [26, Conj. 22]. If moreover Christol's conjecture is true, then there also exist diagonals of infinite grade [22, Prop. 1].

Acknowledgements. We are grateful to Jakob Steininger, who provided a combinatorial idea which led us to the proof of Lemma 1, and to Herwig Hauser for his interest and engaging questions which led to Section 4. Alin Bostan was supported in part by DeRerumNatura ANR-19-CE40-0018 and Sergey Yurkevich was fully supported by the Austrian Science Fund FWF, P-313338.

Proof. By the same argument as in the proof of Lemma 1, the left-hand side is equal to b N k multiplied with the coefficient of

Because the product in the middle does not depend on x N -1 and since we assumed b N -1 +b N = -1, we can first compute

by Lemma 2. Therefore we are left with

which is computed using Lemma 1.

Note that the requirement b N -1 + b N = -1 comes from the +1 in the denominator of the left-hand side in Lemma 2. Since this identity is itself surprising and does not allow for obvious generalizations, the condition on the relationship of b N -1 and b N is necessary.

Proof of Theorem 3

For the proof of Theorem 3 we will only use Lemma 1 and algebraic manipulations similar to the proof of Bober's Lemma 4.1 in [START_REF] Bober | Factorial ratios, hypergeometric series, and a family of step functions[END_REF].

By Lemma 1 we obtain the coefficient of t n for any n ∈ N on the left-hand side:

For the right-hand side we use the fact that for all a, b and non-negative integers n it holds

Then

for all k = 1, . . . , N . Similarly,

Using the same final observation as before we conclude the proof.

Algebraicity and Hadamard grade 4.1 Algebraic cases

We address here the following question: given b

an algebraic function?

In the proof below we will use several times the following useful fact [START_REF] Christol | Diagonals of rational fractions[END_REF]Thm. 33]: if a generalized hypergeometric function is algebraic, then its monodromy weight is zero, that is the number of integer bottom parameters is at most equal to the number of integer top parameters.

Proof. By Theorem 3 it is sufficient to study the algebraicity of the generalized hypergeometric function H(t) defined by

where u k and v k , k = 1, . . . , N -1 are defined like in §3.1:

and

By definition, N -1 of the bottom parameters are ones. We claim that each tuple u k contains at most one integer and if it does contain one, then v k does as well. From the definition of u k it follows that if some u k i ∈ Z then b ∈ Z and b ≡ -i + 1 mod + 1. This shows that for any k = 1, . . . , N -1 at most one u k i ∈ Z. Because of the definition of v k we see that if b ∈ Z and b ≡ i mod for some i ∈ {1, . . . , }, then v k i ∈ Z. This proves the claim.

is a transcendental 2 F 1 , which can be written as the Hadamard product of two algebraic functions

(the 2 F 1 being algebraic by Schwarz's classification [START_REF] Schwarz | Über diejenigen Fälle, in welchen die Gaußische hypergeometrische Reihe einer algebraische Funktion ihres vierten Elementes darstellt[END_REF]) hence its Hadamard grade is 2.

Similarly, the diagonals from ( 2) and ( 3) have Hadamard grade 2 due to the identities

and to the fact that the two 3 F 2 's on the right-hand side are algebraic by the interlacing criterion [3, Thm. 4.8]; see Figure 1 for a pictorial proof, where red points correspond to top parameters, and blue points to bottom parameters (and the additional parameter 1). More generally, the diagonal from (4) has Hadamard grade 2 due to the identity

since the 3 F 2 on the right-hand side is an algebraic function for any R ∈ Q (with Fig. 1 replaced by a similar one, containing only interlacing blue right triangles and red equilateral triangles). This observation provides an alternative (and probably the shortest) proof that the hypergeometric functions in (2), ( 3) and ( 4) are diagonals of algebraic functions.

The same observation also quickly solves two more cases amongst the 16 cases in the list [5, p. 58