Philippe Biane

Orthogonal polynomials on the unit circle, q-Gamma weights, and discrete Painlevé equations

Keywords: 1991 Mathematics Subject Classification, Primary ; Secondary, 1

We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of q-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlevé equations, in a Lax form, which correspond to an A

(1) 3 surface in Sakai's classification.

Introduction

Motivated by number theoretic considerations, J.F. Burnol [START_REF] Burnol | Scattering on the p-adic field and a trace formula[END_REF], [START_REF] Burnol | Des équations de Dirac et de Schrödinger pour la transformation de Fourier[END_REF], has raised the problem of realizing the Fourier transform as a scattering. More precisely, consider the Fourier transform restricted to even functions on R which is given, after appropriate normalization, by The interaction between the additive Fourier transform and the Mellin transform is captured by the formula, where the Fourier transform acts on tempered distributions,

(1.1) Fx +ik = π -ik Γ(1/4 -ik/2) Γ(1/4 + ik/2) x -ik k real which yields the gamma factor coming from the place at infinity in the functional equation for Riemann zeta function. Gamma factors for other L-functions are obtained by similar considerations involving the radial Fourier transform in R n . Interpreting x ik as an incoming wave, and x -ik as the corresponding outgoing wave (in logarithmic coordinates) on the space ]0, +∞[, it is a natural question to ask what is the scattering potential which realizes the map (1.1) (we shall recall the basics of scattering theory in the next section). This was answered by Burnol [6] who showed that, more generally, the function (1.2) Γ(a -ik/2)Γ(b + ik/2) Γ(a + ik/2)Γ(b -ik/2) (a, b real, a > b), is the scattering phase shift for a Dirac equation on ]0, +∞[, where the potential is given in terms of Painlevé transcendents. In this paper we consider a discrete analogue of this problem. Discrete scattering problems have been introduced by Case and Kac [START_REF] Case | A discrete version of the inverse scattering problem[END_REF] and studied further by Case and Geronimo [START_REF] Geronimo | Scattering theory and polynomials orthogonal on the unit circle[END_REF]. In this last paper it is shown that, given a function f , analytic inside the unit disk, continuous on the unit circle, then f (e iθ )/f (e iθ ) is the phase shift for a discrete scattering problem, and solving the inverse problem for this scattering problem is equivalent to finding the recursion relations for the orthogonal polynomials on the unit circle with respect to the measure dθ/|f (e iθ )| 2 . Recall that, if φ n is the sequence of these polynomials, and

φ * n (z) = z n φn (1/z)
then the recursion relation, due to Szegö, can be written, with

Φ n (z) = φ n (z) φ * n (z) as (1.3) Φ n+1 (z) = B n (z)Φ n (z)
where

(1.4) B n (z) = z α n+1 z ᾱn+1 1 .
The α n are the Verblunsky coefficients of the sequence φ n , see e.g. [START_REF] Szegö | Orthogonal polynomials[END_REF], [START_REF] Geronimus | Polynomials orthogonal on a circle and interval[END_REF], or [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 1. Classical theory[END_REF], [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 2. Spectral theory[END_REF]. Equation (1.3) is a discrete analogue of a Dirac system leading to scattering. Thus a natural problem, in view of Burnol's question, is to find the recursion coefficients for a weight w, expressed in terms of q-Gamma functions, of the form w(z) = (az, q) ∞ (āz -1 , q) ∞ (bz, q) ∞ ( bz -1 , q) ∞ .

with, as usual,

(z, q) ∞ = ∞ 0 (1 -zq i ).
Indeed w(e iθ ) = 1/|f (e iθ )| 2 with f (e iθ ) = (be iθ , q) ∞ /(ae iθ , q) ∞ , and taking a suitable limit of f (e iθ )/f (e iθ ) for q → 1 yields the phase shift (1.2). The weight w satisfies a q-difference equation

w(qz) = V (z) W (z) w(z)
where V and W are two polynomials of degree 2. It turns out that a method, which goes back to Laguerre, further used by Freud [START_REF] Freud | On the coefficients in the recursion formulae of orthogonal polynomials[END_REF], allows, in this situation, to obtain nonlinear relations between these recursion coefficients. This kind of question has seen renewed interest recently, see e.g. [START_REF] Magnus | Freud's equations for orthogonal polynomials as discrete Painlevé equations[END_REF], [START_REF] Forrester | Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N -recurrences for averages over U (N )-P III and P V τ -functions[END_REF], [START_REF] Van Assche | Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials[END_REF]. We shall prove that

Φ n (qz) = 1 V (z) A n (z)Φ n (z)
where A n is a matrix with polynomial entries, of degree 2, which can be computed from the coefficients of the orthogonal polynomials. Compatibility between this equation and (1.3) implies the relation

A n+1 (z)B n (z) = B n (qz)A n (z).
This last equation can be interpreted as an isomonodromy deformation of the q difference operator associated with the matrix A n . It is similar, although it is not equivalent to, the Lax form of the discrete Painlevé VI equation studied by Jimbo and Sakai in [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF]. We shall study it in details and prove that the equation that we obtain in this way is a Lax form for a discrete Painlevé equation, corresponding to the A

3 surface, according to the classification of Sakai [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]. As we shall see, this equation looks like the one of [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF], but is obtained from translation in a different direction in the affine Coxeter group D

(1) 5 . This paper is organized as follows. We start by recalling some basic material on one-dimensional scattering theory, following [START_REF] Faddeev | The inverse problem in the quantum theory of scattering[END_REF], for the continuous case, and [START_REF] Geronimo | Scattering theory and polynomials orthogonal on the unit circle[END_REF] for the discrete case. We also recall some of the theory of orthogonal polynomials on the unit circle, following closely [START_REF] Forrester | Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N -recurrences for averages over U (N )-P III and P V τ -functions[END_REF]. Laguerre's method is used in section 3 to derive nonlinear equations satisfied by the recursion coefficients of these orthogonal polynomials. Then, in section 4, we give an explicit formula for these nonlinear relations as a birational map in two variables, and show that they give rise to a regular isomorphism between two surfaces obtained by blowing up P 1 × P 1 at eight points. In section 5 we recall Sakai's theory, and use it to identify our nonlinear relation as a discrete Painlevé equation. We determine its type in Sakai's classification, and find its expression as a certain translation in the affine Weyl group of type D

(1) 5 . Finally in the last section we derive a differential system by taking a continuous limit of the discrete equations.

The results of this paper are based on numerous lengthy computations. In order not to bore the reader we have omitted the details of most of these computations, but given enough details so that they can be reproduced.

I would like to thank Jean-François Burnol for keeping me informed about his work which motivated me to undertake the study of these equations. I thank also Alexei Borodin and Philip Boalch for their useful comments at an early stage of this work.

2. One-dimensional scattering, continuous and discrete 2.1. Scattering for the Schrödinger equation. One starts with the equation

(2.1) - d 2 dx 2 ψ(x, s) + q(x)ψ(x, s) = s 2 ψ(x, s) s ∈ C
on the half line [0, +∞[. This equation has a unique solution, ψ(x, s), satisfying the boundary condition

(2.2) ψ(0, s) = 0, d dx ψ(0, s) = 1.
For a potential q decreasing sufficiently at infinity there are also two solutions, the Jost solutions f (x, ±s), determined by the conditions

f (x, ±s)e ∓isx → x→∞ 1. (2.3)
For real k, the Jost solutions f (x, k) and f (x, -k) correspond to incoming and outgoing waves and ψ is a linear combination

(2.4) ψ(x, k) = 1 2ik f (x, k) M (k) -f (x, -k)M (k)
such that M is the boundary value of the function M (s) = f (0, s), analytic in the upper half-plane. It follows that, as x → ∞,

ψ(x, k) = |M (k)| k sin(kx -η(k)) + o(1)
where η(k) = arg M (k) is the phase shift created by the potential q, with respect to the case q = 0. The problem of inverse scattering is, given the phase shift η, to reconstruct the potential q. For phase shifts satisfying a growth condition at infinity, there exists a unique solution without bound states, which can be recovered by the methods developped in the 50's, see e.g. [START_REF] Gelfand | On the determination of a differential equation from its spectral function[END_REF] and [START_REF] Faddeev | The inverse problem in the quantum theory of scattering[END_REF].

Discrete scattering.

We now follow the exposition of Case and Geronimo [START_REF] Geronimo | Scattering theory and polynomials orthogonal on the unit circle[END_REF]. More information on orthogonal polynomials on the unit circle can be found e.g. in Szegö [START_REF] Szegö | Orthogonal polynomials[END_REF], Geronimus [START_REF] Geronimus | Polynomials orthogonal on a circle and interval[END_REF], or the comprehensive treatise by Simon [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 1. Classical theory[END_REF], [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 2. Spectral theory[END_REF].

Let µ be a finite positive measure on the unit circle U (1). Denote its moments The α n are the Verblunsky coefficients of the system, cf [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 1. Classical theory[END_REF]. They satisfy

σ n+1 = (1 -|α n+1 | 2 )σ n
thus the sequence σ n is determined by σ 0 = µ(U (1)) and the sequence α n . The orthogonal polynomials are subject to the recursion relation

(2.5)    φ n+1 (z) = zφ n (z) + α n+1 z n φn (1/z) φn+1 (1/z) = z -n φ n (z) + ᾱn+1 z -1 φn (1/z)
In matrix form, putting

Ψ(z, n) = φ n (z) φn (1/z)
and

D(n) = z α n+1 z n ᾱn+1 z -n z -1 one can write Ψ(z, n + 1) = D(n)Ψ(z, n)
Assuming that the potential α satisfies lim n→∞ α n = 0 fast enough, one can construct the Jost solutions to equation (2.5)

Ψ + (z, n) = φ + z, n) φ+ (z, n) Ψ -(z, n) = φ-(z, n) φ -(z, n)
satisfying the boundary conditions

lim n |φ + (z, n) -z n | = 0 |z| > 1 lim n |φ -(z, n) -z -n | = 0 |z| < 1 lim n | φ+ (z, n)| = 0 |z| > 1 lim n | φ-(z, n)| = 0 |z| < 1.
The solutions Ψ ± are linearly independent, and

(2.6) Ψ(z, n) = f + (z)Ψ + (z, n) + f -(z)Ψ -(z, n) where f + (z) = f-(z) for |z| = 1 Equation (2.6
) is the discrete analogue of (2.4).

The function f + has analytic continuation inside the unit disc, it is related to the original measure µ on the unit circle by

dµ(θ) = dθ |f + (e iθ )| 2
The phase η(θ) = arg(f + (e iθ )) is the phase shift induced by the potential α n . It can be recovered from the function 1 |f+(e iθ )| 2 on the unit circle. One sees that reconstructing the phase shift from the potential α is equivalent to finding the orthogonality relation for the sequence of polynomials φ n , solution to the recursion equation (2.5). Conversely, the inverse scattering problem is the easy problem of finding the recursion coefficients for the orthogonal polynomials, knowing their orthogonality measure. In fact the connection between scattering theory and orthogonal polynomials has been known for a long time and is at the heart of the Gelfand-Levitan method (see [START_REF] Gelfand | On the determination of a differential equation from its spectral function[END_REF]).

2.3.

Comparing the discrete and continuous cases. Given a solution A of the Riccatti equation q = A + A 2 the equation (2.1), for s = k real, is equivalent to the Dirac system (2.7) dy/dx = Ay + kz, -dz/dx = Az + ky.

The Dirac system (2.7) can then be put in the form

(2.8) dW/dx = A W e 2ikx , d W /dx = AW e -2ikx .
with W = (y + iz)e ikx Similarly, putting ω n (θ) = e -inθ φ n (e iθ ) the system (2.5) becomes (2.9) ω n+1 -ω n = α n+1 e -inθ ωn , ωn+1 -ωn = ᾱn+1 e inθ ω n .

It is now clear that (2.9) is a discretized version of (2.8).

2.4. Recursion relations and the Caratheodory function. We recall a few properties of the orthogonal polynomials which will be needed in the sequel.

The function

F (z) = e iθ + z e iθ -z dµ(θ)
is called the Caratheodory function of µ, it is holomorphic in C * \ U (1), and has the expansion

F (z) = c 0 + 2 ∞ k=1 c k z k |z| < 1 F (z) = -c 0 -2 ∞ k=1 c -k z -k |z| > 1 Let φ * n (z) = z n φn (1/z)
and introduce the associated polynomials

ψ n (z) = e iθ + z e iθ -z (φ n (e iθ ) -φ n (z))dµ(θ) ψ 0 (z) = 1 ψ * n (z) = - e iθ + z e iθ -z (z n φ n (e iθ ) -φ * n (z))dµ(θ)
ψ * 0 (z) = 1 and the functions

n (z) = ψ n (z) + F (z)φ n (z) = e iθ + z e iθ -z φ n (e iθ )dµ(θ) * n (z) = ψ * n (z) -F (z)φ * n (z) = e iθ + z e iθ -z φ(e iθ )z n dµ(θ) then, for n ≥ 1, n (z) = 2σ n z n + O(z n+1 ) z → 0 (2.10) n (z) = 2σ n α n+1 z -1 + O(z -2 ) z → ∞ (2.11) * n (z) = 2σ n ᾱn+1 z n+1 + O(z n+2 ) z → 0 (2.12) * n (z) = 2σ n + O(z -1 ) z → ∞ (2.13)
In other words, the fractions -ψn φn and

ψ * n φ * n
are reduced continued fractions for F , near ∞ and 0 respectively. Furthermore,

n+1 (z) = z n (z) -α n+1 * n (z) * n+1 (z) = * n (z) -ᾱn+1 z n (z).
The recursion relations can be put in the form

(2.14) Y n (z) = φ n (z) n(z) w(z) φ * n (z) - * n (z) w(z) (2.15) Y n+1 (z) = B n (z)Y n (z). with (2.16) B n (z) = z α n+1 z ᾱn+1 1
There are also discrete Wronskian identities, derived from the recursion relations, namely

φ n+1 (z)ψ n (z) -ψ n+1 (z)φ n (z) = φ n+1 (z) n (z) -n+1 (z)φ n (z) (2.17) = 2α n+1 σ n z n φ * n+1 (z)ψ * n (z) -ψ * n+1 (z)φ * n (z) = φ * n+1 (z) * n (z) - * n+1 (z)φ * n (z) (2.18) = 2ᾱ n+1 σ n z n+1 φ n (z)ψ * n (z) + ψ n (z)φ * n (z) = φ n (z) * n (z) + n (z)φ * n (z) (2.19) = 2σ n z n .
3. Nonlinear relations for the Verblunsky coefficients 3.1. Laguerre's method. We now adapt the method of Laguerre [START_REF] Laguerre | Sur la réduction en fractions continues d'une fonction qui satisfait à une équation linéaire du premier ordre à coefficients rationnels[END_REF] and Freud [START_REF] Freud | On the coefficients in the recursion formulae of orthogonal polynomials[END_REF] for finding nonlinear relations between the Verblunsky coefficients. The computations that follow are inspired by Forrester and Witte [START_REF] Forrester | Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N -recurrences for averages over U (N )-P III and P V τ -functions[END_REF], who treat the case of difference equations of the form w(z + 1) = ρ(z)w(z). Another derivation for the q-recurrence relations can be found in the paper [START_REF] Ismail | Discriminants and functional equations for polynomials orthogonal on the unit circle[END_REF] by Ismail and Witte, however we shall need them in the form below, which is slightly different from the results of [START_REF] Ismail | Discriminants and functional equations for polynomials orthogonal on the unit circle[END_REF], so we give a detailed derivation here. Let us assume that the measure µ has the form dµ(θ) = w(e iθ )dθ with w holomorphic in C * , satisfying

(3.1) w(qz) = ρ(z)w(z)
where q ∈]0, 1[, and ρ is a rational function,

(3.2) ρ(z) = V (z) W (z)
for some polynomials V, W . The case of interest is when

(3.3) w(e iθ ) = (ae iθ , q) ∞ (be iθ , q) ∞ 2 = (ae iθ , q) ∞ (āe -iθ , q) ∞ (be iθ , q) ∞ ( be -iθ , q) ∞ q ∈]0, 1[, a, b ∈ C * , thus w(z) = (az, q) ∞ (āz -1 , q) ∞ (bz, q) ∞ ( bz -1 , q) ∞ satisfies w(qz) = (qz -ā)(1 -bz) (qz -b)(az -1) w(z),
and we may take

V (z) = (qz -ā)(bz -1) (3.4) W (z) = (qz -b)(1 -az). (3.5)
For the moment we shall make the computations with general polynomials V and W , and come back to the specific values (3.4), (3.5) in the last section.

Equation (3.1) implies that the Caratheodory function satisfies

(3.6) W (z)F (qz) = V (z)F (z) + U (z)
with U a polynomial of degree at most D = max(deg W, deg V ). Indeed, take the partial fraction expansion

ρ(z) = δ(z) + j ω j z -a j , δ polynomial of degree less than (deg V -deg W ) + with W (z) = C j (z -a j ), then F (qz) = ζ+qz ζ-qz w(ζ) dζ iζ = ζ+z ζ-z w(qζ) dζ iζ by a change of integration contour = ζ+z ζ-z ρ(ζ)w(ζ) dζ iζ = ρ(z)F (z) + ζ+z ζ-z (ρ(ζ) -ρ(z))w(ζ) dζ iζ = ρ(z)F (z) + δ(ζ, z) + j ωj z-aj ζ+z ζ-aj w(ζ) dζ iζ δ(ζ, z) = (ζ + z) δ(ζ)-δ(z) ζ-z = (V (z)F (z) + U (z))/W (z)
where U is a polynomial of degree at most D. We shall now use (3.6) and the characterization of the quotients ψ n /φ n and ψ * n /φ * n as reduced continued fraction expansions. Let us start from

n (z) = ψ n (z) + F (z)φ n (z) and n (qz) = ψ n (qz) + ( V (z) W (z) F (z) + U (z) W (z) )φ n (qz) by (3.6)
and eliminate F (z) between these two equations, to get

W (z) n (qz)φ n (z) -V (z) n (z)φ n (qz) = W (z)ψ n (qz)φ n (z) -V (z)ψ n (z)φ n (qz) + U (z)φ n (z)φ n (qz).
The right hand side is a polynomial in z and, according to (2.10)-(2.13), the left hand side is O(z n ) near 0 and O(z D+n-1 ) near ∞, therefore there exists a polynomial Θ n (z) of degree less than D -1, such that

(3.7) 2α n+1 σ n z n Θ n (z) = W (z) n (qz)φ n (z) -V (z) n (z)φ n (qz).
In case α n+1 = 0, one has φ n+1 (z) = zφ n (z) and φ * n+1 (z) = φ * n (z), and the sought relation for φ n+1 (qz) can be obtained from that of φ n (qz). From

2α n+1 σ n z n Θ n (z) = W (z)ψ n (qz)φ n (z) -V (z)ψ n (z)φ n (qz) + U (z)φ n (z)φ n (qz)
we get, according to (2.17),

[φ n+1 (z)ψ n (z) -ψ n+1 (z)φ n (z)]Θ n (z) = W (z)ψ n (qz)φ n (z) -V (z)ψ n (z)φ n (qz) +U (z)φ n (z)φ n (qz) or ψ n (z)(φ n+1 (z)Θ n (z)+V (z)φ n (qz)) = φ n (z)(W (z)ψ n (qz)+U (z)φ n (qz)+ψ n+1 Θ n (z)).
This expression is a polynomial, common multiple of φ n and ψ n which have no common zero (see (2.17)), so it can be put in the form

Ω n (z)φ n (z)ψ n (z)
where Ω n is a polynomial of degree at most D. Thus Ω n satisfies

Ω n (z)φ n (z) = φ n+1 (z)Θ n (z) + V (z)φ n (qz) (3.8) Ω n (z)ψ n (z) = W (z)ψ n (qz) + U (z)φ n (qz) + ψ n+1 (z)Θ n (z) (3.9)
One obtains Ω n by multiplying (3.9) by φ n+1 , substracting (3.8) multiplied by ψ n+1 , and using (2.17), thus

(3.10) 2α n+1 σ n z n Ω n (z) = W (z) n (qz)φ n+1 (z) -V (z) n+1 (z)φ n (qz) Equation (3.8) gives (3.11) V (z)φ n (qz) = Ω n (z)φ n (z) -φ n+1 (z)Θ n (z)
or, using (2.5), (3.12) (3.8) by F (z) and adding (3.9)

V (z)φ n (qz) = (Ω n (z) -zΘ n (z))φ n (z) -α n+1 Θ n (z)φ * n (z) On multiplying
Ω n (z) n (z) = V (z) n (qz) w(z) w(qz) + n+1 (z)Θ n (z) therefore Ω n (z) n (z) w(z) = V (z) n (qz)) w(qz) + n+1 (z) w(z) Θ n (z). Finally (3.13) V (z) n (qz)) w(qz) = Ω n (z) n (z) w(z) - n+1 (z) w(z) Θ n (z) and (3.14) V (z) n (qz)) w(qz) = (Ω n (z) -zΘ n (z)) n (z) w(z) + α n+1 Θ n (z) * n (z) w(z)
Equations for φ * n and * n are obtained by similar arguments. One finds polynomials Θ * n (z) and Ω * n (z) with the same bounds on their degrees as Θ n (z) and Ω n (z), such that

(3.15) V (z)φ * n (qz) = (Ω * n (z) -Θ * n (z))φ * n (z) -z ᾱn+1 Θ * n (z)φ n (z) and (3.16) V (z) * n (qz) w(qz) = (Ω * n (z) -Θ * n (z)) * n (z) w(z) + z ᾱn+1 Θ * n (z) n (z) w(z) Proposition 3.1. Let (3.17) A n (z) = Ω n (z) -zΘ n (z) α n+1 Θ n (z) -z ᾱn+1 Θ * n (z) Ω * n (z) -Θ * n (z)
.

then, with Y n defined by (2.14), the relations (3.12), (3.14), (3.15), (3.16) can be put in matrix form as

(3.18) Y n (qz) = 1 V (z) A n (z)Y n (z)
Compatibility relations between (2.15) and (3.18) imply that

(3.19) A n+1 (z)B n (z) = B n (qz)A n (z).
The coefficients of the polynomials Θ n , Θ * n , Ω n , Ω * n in (3.17) can be computed from the coefficients of the polynomials φ n . For example, with V and W given by (3.4) and (3.5) one finds, putting

Θ n (z) = λz + µ (3.20) 2α n+1 z n Θ n (z) = W (z) n (qz)φ n (z) -V (z) n (z)φ n (qz)
and comparing the two expressions near zero,

2α n+1 z n µ = 2α n z n ( bq n -ā)
Comparing at infinity gives λ, and finally

(3.21) Θ n (z) = (a -bq n+1 )z + (-ā + bq n ) α n α n+1 .
Similar reasoning yield the other polynomials in terms of the Verblunsky coefficients. Using

φ n (z) = z n + β n z n-1 + . . . + α n with β n = n j=1 α j ᾱj-1 One finds (3.22) Ω n (z) -zΘ n (z) = bq n+1 z 2 + tz + bq n
where t is a complicated expression in terms of α n , α n+1 and β n . Similarly [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 1. Classical theory[END_REF]) implies nonlinear recursion relations among these coefficients.

Θ * n (z) = (aq -bq n+1 ) ᾱn ᾱn+1 z + bq n+1 -ā (3.23) Ω * n (z) -Θ * n (z) = aqz 2 + t * z + ā (3.24) Equation (3.

The isomonodromy deformation.

Rather than do the tedious computations here, we shall, in the next section, start from the relation (3.19) and derive everything from its qualitative features. This will allow us to make the connection with Sakai's theory of discrete Painlevé equations. Before this we first make a few remarks. Equation (3.19) has the form

(3.25) Ã(z) = B(qz)A(z)B(z) -1
where

A(z) = A d z d + . . . + A 0 , Ã(z) = Ãd z d + . . . + Ã0
and

B(z) = B z 0 0 1
for some constant matrix B. Equation (3.25) is an isomonodromy transformation of a q-difference equation, as studied by Birkhoff [START_REF] Birkhoff | General theory of linear difference equations[END_REF], and recently by Borodin [START_REF] Borodin | Isomonodromy transformations of linear systems of difference equations[END_REF]. It is closely related to the equation appearing in [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF]. Such isomonodromy transformations, for difference equations instead of q-difference equations, have been studied in depth by Arinkin and Borodin in [START_REF] Arinkin | Moduli spaces of d-connections and difference Painlevé equations[END_REF], using geometric methods. These methods, involving the definition of "q-connections" should also apply to our situation, although we do not use them here.

In our case, the matrices A d , Ãd are lower triangular matrices, and A 0 , Ã0 are upper triangular. Let κ 1 , κ 2 be the eigenvalues of A d , and θ 1 , θ 2 those of A 0 , so that

A d = κ 1 0 * κ 2 , A 0 = θ 1 * 0 θ 2
Consider the matrix

Â(z) = qz 0 0 1 A(z) z -1 0 0 1 = Â2 z 2 + Â1 z + Â0 then Âd = qκ 1 * 0 κ 2 , Â0 = qθ 1 0 * θ 2 , det Â(z) = qdetA(z)
We can now conjugate Â(z) by a constant matrix B in order to get the matrix Ã(z) with

Ãd = qκ 1 0 * κ 2 , Ã0 = qθ 1 * 0 θ 2 .
The matrix B is determined up to left multiplication by a diagonal invertible matrix. One can choose its coefficients to be expressed as rational functions of the coefficients of A(z). Furthermore equation (3.25) can be used in the same way to deduce A(z) from Ã(z). It follows that equation (3.25) defines a birational transformation between two spaces of matrices with polynomial coefficients. We shall give an explicit formula for this transformation in the next section. The recursion equations for Verblunsky coefficients are given by iterating this transformation. In the next section we shall investigate in details the case where d = 2, which corresponds to the case of the Verblunsky coefficients for the q-Gamma weight (3.3).

A birational transformation

4.1. The space X κ1,κ2,θ1,θ2,c1,c2,c3,c4 . We now embark on the analysis of equation (3.25), in the case of degree 2 polynomial matrices. Consider a two-dimensional vector space H, and let

A(z) = A 0 + A 1 z + A 2 z 2
be an endomorphism of H which depends polynomially on the variable z, with A 0 , A 1 , A 2 , elements of End(H).

Definition 4.1. We let Xκ1,κ2,θ1,θ2,c1,c2,c3,c4

denote the space of all such A(z) satisfying

{κ 1 , κ 2 } = sp A 2 , {θ 1 , θ 2 } = sp A 0 and (4.1) det A(z) = κ 1 κ 2 (z -c 1 )(z -c 2 )(z -c 3 )(z -c 4 )
for some fixed complex constants κ 1 , κ 2 , θ 1 , θ 2 , c 1 , . . . , c 4 , satisfying the equation

(4.2) κ 1 κ 2 c 1 c 2 c 3 c 4 = θ 1 θ 2
(which comes from considering det A(0)). We let

X κ1,κ2,θ1,θ2,c1,c2,c3,c4
denote its quotient by the action of GL(H) by conjugation. These spaces are complex manifolds.

In the sequel c 1 , . . . , c 4 will be fixed so we will omit them from the notation most of the time.

We consider the case of generic parameters, in particular κ 1 = κ 2 , θ 1 = θ 2 , and A 0 , A 2 are semisimple. We denote E 1 , E 2 the eigenspaces of A 2 , and F 1 , F 2 those of A 0 . The set of A(z) such that E 2 = F 1 is a dense open set Ω, invariant by conjugation, and for each element of this open set we can chose a basis (u, v) with u ∈ F 1 , v ∈ E 2 so that, in this basis, A 0 and A 2 have matrices of the form

A 2 = κ 1 0 * κ 2 , A 0 = θ 1 * 0 θ 2
As in the preceding section, let q be a nonzero complex number and consider the matrix

Â(z) = qz 0 0 1 A(z) z -1 0 0 1 = Â2 z 2 + Â1 z + Â0 then Â2 = qκ 1 * 0 κ 2 , Â0 = qθ 1 0 * θ 2 , det Â(z) = qdetA(z)
therefore the matrix Â(z) belongs to the space Xqκ1,κ2,qθ1,θ2 . Thus A(z) → Â(z) defines a map from a dense open subset of Xκ1,κ2,θ1,θ2 to Xqκ1,κ2,qθ1,θ2 . Furthermore, as explained also at the end of the preceding section, this map can be inverted, thus defining a bijection between dense open subsets of Xκ1,κ2,θ1,θ2 and Xqκ1,κ2,qθ1,θ2 .

Clearly the map passes to the quotients and induces a bijection between dense open subsets of X κ1,κ2,θ1,θ2 and X qκ1,κ2,qθ1,θ2 , which is a birational map.

Definition 4.2. We call φ : X κ1,κ2,θ1,θ2 → X qκ1,κ2,qθ1,θ2 the birational map constructed above.

In the next section, we shall introduce a parametrization of the spaces in order to give an explicit formula for this birational transformation. 4.2. Jimbo and Sakai parametrization. We will use the following parametrization of the space X κ1,κ2,θ1,θ2 , after Jimbo and Sakai [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF] (4.3)

A(z) = κ 1 ((z -y)(z -α) + z 1 ) z -y z(γz + β) κ 2 ((z -y)(z -δ) + z 2 )
Thus y is the unique root of the polynomial which is the 12 coefficient of A, and κ 1 z 1 , κ 2 z 2 are the values of the 11, resp. 22 coefficient of A(z) at z = y. One has

yα + z 1 = θ 1 /κ 1 , yα + z 2 = θ 2 /κ 2
The quantities y, z 1 , z 2 satisfy (4.4)

z 1 z 2 = (y -c 1 )(y -c 2 )(y -c 3 )(y -c 4 )
where the equality follows from considering the determinant of A(y). Let us put

ξ = (y -c 1 )(y -c 2 )/κ 1 z 1 = κ 2 z 2 /(y -c 3 )(y -c 4 ).
The quantities β and γ are determined as rational functions of y, ξ using (4.1). It follows that the variables (y, ξ) in C * × C * parametrize a dense open subset of X κ1,κ2,θ1,θ2 . Using this parametrization, the space X κ1,κ2,θ1,θ2 is identified with P 1 × P 1 blown up at the seven points (c 1 , 0), (c 2 , 0), (c 3 , ∞), (c 4 , ∞) (0, c 1 c 2 /θ 1 ), (0, c 1 c 2 /θ 2 ), (∞, 1/κ 1 ) minus the strict transforms of the four lines y = 0, y = ∞, ξ = 0, ξ = ∞, see Fig. 2 below.

(c 1 , 0) (c 2 , 0) (c 3 , ∞) (c 4 , ∞) (0, c 1 c 2 /θ 1 ) (0, c 1 c 2 /θ 2 ) (∞, 1/κ 1 ) y = 0 ξ = 0 ξ = ∞ y = ∞ Fig. 2. The space X κ1
The curves corresponding to F 2 = E 1 and F 2 = E 2 are respectively the strict transforms of the points (0, c 1 c 2 /θ 2 ) and (0, c 1 c 2 /θ 1 ).

Explicit birational expression for φ.

We now describe the map φ as a birational transformation, using the Jimbo-Sakai coordinates (4.3). Let A(z) be as in (4.3) and consider the matrix

Â(z) = qz 0 0 1 A(z) z -1 0 0 1 = qκ 1 ((z -y)(z -α) + z 1 ) qz(z -y) (γz + β) κ 2 ((z -y)(z -δ) + z 2 )
Then Â(z) belongs to the space X qκ1,κ2,qθ1,θ2 . In order to find its Jimbo-Sakai coordinates, we have to use the basis formed by the spaces Ê2 and F1 . It follows that Ã

(z) = B -1 Â(z)B with B = qθ 1 -θ 2 -q β qκ 1 -κ 2
on the condition that the matrix B so defined is invertible. In order to go from à to A one proceeds in the opposite way. Start from

à = qκ 1 ((z -ỹ)(z -α) + z1 ) z - ỹ z(γz + β) κ 2 ((z -ỹ)(z -δ) + z2 )
conjugate it by a constant matrix, so that its 21 coefficient has degree 1 and its 12 coefficient is z×(a polynomial with degree 1). For this use eigenvectors of Ã2 and Ã0 , thus Â(z) = C -1 Ã(z)C with

C = qκ 1 -κ 2 -ỹ qγ qθ 1 -θ 2
We will not compute the inverse transformation.

Let us now compute an explicit formula for φ, using the coordinates (y, ξ). We start from the matrix

A(z) = κ 1 z 2 + az + θ 1 z -y z(γz + β) κ 2 z 2 + bz + θ 2 with a = -κ 1 y + (y -c 1 )(y -c 2 ) yξ - θ 1 y (4.5) b = -κ 2 y + κ 1 κ 2 ξ(y -c 3 )(y -c 4 ) y - θ 2 y (4.6) and θ 1 b + θ 2 a + βy = -κ 1 κ 2 σ 3 (4.7) κ 1 b + κ 2 a -γ = -κ 1 κ 2 σ 1 (4.8)
where the σ i are the elementary symmetric functions in the c i . Then (4.9) Ã(z) = B(qz)A(z)B(z

) -1 = qκ 1 z 2 + ãz + qθ 1 w(z -ỹ) w -1 z(γz + β) κ 2 z 2 + bz + θ 2 with B(z) = z(qκ 1 -κ 2 ) q -zβ qθ 1 -θ 2 B(z) -1 = ∆ -1 z -1 (qθ 1 -θ 2 ) -z -1 q β qκ 1 -κ 2 ∆ = (qκ 1 -κ 2 )(qθ 1 -θ 2 ) + qβ
A computation using (4.9) gives wỹ = q, and

w = R/∆, ỹ = q∆ R with ξy 2 ∆ = ξy 2 (qθ 1 -θ 2 )(qκ 1 -κ 2 ) + qξy 2 β = ξy 2 (qθ 1 -θ 2 )(qκ 1 -κ 2 ) +qyξ(-κ 1 κ 2 σ 3 -θ 1 b -θ 2 a) using(4.7) = -qκ 1 κ 2 θ 1 ξ 2 (y -c 3 )(y -c 4 ) +ξ((q 2 κ 1 θ 1 + κ 2 θ 2 )y 2 -qκ 1 κ 2 σ 3 y + 2qθ 1 θ 2 )
-qθ 2 (y -c 1 )(y -c 2 ) using (4.5), (4.6), and

ξyR = ξy((qκ 1 -κ 2 )(-aq 2 + bq -q(qκ 1 -κ 2 )y) -q 2 γ) = ξy(-q 3 κ 1 a -qκ 2 b -q(qκ 1 -κ 2 ) 2 y -q 2 κ 1 κ 2 σ 1 ) by (4.8) = q 3 κ 1 (κ 1 ξy 2 -(y -c 1 )(y -c 2 ) + θ 1 ξ) +qκ 2 (κ 2 ξy 2 -κ 1 κ 2 ξ 2 (y -c 3 )(y -c 4 ) + θ 2 ξ) -q(qκ 1 -κ 2 ) 2 y 2 ξ -q 2 κ 1 κ 2 σ 1 ξy by (4.5), (4.6) = -qκ 1 κ 2 2 ξ 2 (y -c 3 )(y -c 4 ) +ξ(2q 2 κ 1 κ 2 y 2 -q 2 κ 1 κ 2 σ 1 y + q(q 2 κ 1 θ 1 + κ 2 θ 2 )) -q 3 κ 1 (y -c 1 )(y -c 2 ). Therefore ỹ = S yT (4.10) with S = -qκ 1 κ 2 θ 1 ξ 2 (y -c 3 )(y -c 4 ) +ξ((q 2 κ 1 θ 1 + κ 2 θ 2 )y 2 -qκ 1 κ 2 σ 3 y + 2qθ 1 θ 2 ) -qθ 2 (y -c 1 )(y -c 2 ) T = -κ 1 κ 2 2 ξ 2 (y -c 3 )(y -c 4 ) +ξ(2qκ 1 κ 2 y 2 -qκ 1 κ 2 σ 1 y + (q 2 κ 1 θ 1 + κ 2 θ 2 )) -q 2 κ 1 (y -c 1 )(y -c 2 ).
In order to find ξ we proceed as follows. From the definition one has ξ = (ỹ -c 1 )(ỹ -c 2 ) qκ 1 ỹ2 + ãỹ + qθ 1 where ã can be computed from (4.9). Plugging in the values of a, b, β, γ, and ỹ given by (4.5), (4.6), (4.7) (4.8), (4.10) gives an unwieldy expression as a rational fraction in the variables y, ξ. One can simplify this expression using the following observation. Computing the determinant det Ã(ỹ) = q det A(ỹ) from (4.9) gives the identity (4.11)

qκ 1 κ 2 (ỹ -c 1 )(ỹ -c 2 )(ỹ -c 3 )(ỹ -c 4 ) = (qκ 1 ỹ2 + ãỹ + qθ 1 )(κ 2 ỹ2 + bỹ + θ 2 )
where ỹ, ã, b are rational expressions in y, ξ. Of course this identity is compatible with the alternative expression

ξ = κ 2 ỹ2 + bỹ + θ 2 qκ 1 κ 2 (ỹ -c 3 )(ỹ -c 4 )
.

Replacing ỹ, ã, b in (4.11) by their values and reducing the denominators, we get a polynomial identity between the numerators. This leads us to expect that the numerator of each expression ỹ -c i factorizes into a product ot two terms, which divide respectively the numerators of qκ 1 ỹ2 + ãỹ + qθ 1 and κ 2 ỹ2 + bỹ + θ 2 . Since the degree of the numerator of ỹ -c i in the variable ξ is two, we expect a factorization into two terms of degree 1 in ξ. Indeed a direct computation confirms that

S -c 1 yT = κ 1 κ 2 ξ 2 (c 1 yκ 2 -qθ 1 )(y -c 3 )(y -c 4 ) +ξ -2qc 1 κ 1 κ 2 y 3 + (q 2 κ 1 θ 1 + κ 2 θ 2 + qc 1 κ 1 κ 2 σ 1 )y 2 +ξ (-qκ 1 κ 2 σ 3 + c 1 (q 2 κ 1 θ 1 + κ 2 θ 2 ))y + 2qθ 1 θ 2 +q(qc 1 κ 1 y -θ 2 )(y -c 1 )(y -c 2 ) = (ξ(c 1 yκ 2 -qθ 1 ) -qc 1 (y -c 2 )) × ×(κ 1 κ 2 ξ(y -c 3 )(y -c 4 ) - 1 c 1 (qc 1 κ 1 y -θ 2 )(y -c 1 ))
and similar expressions for the other c i , e.g.

S -c 3 yT = κ 1 κ 2 ξ 2 (c 3 yκ 2 -qθ 1 )(y -c 3 )(y -c 4 ) +ξ -2qc 3 κ 1 κ 2 y 3 + (q 2 κ 1 θ 1 + κ 2 θ 2 + qc 3 κ 1 κ 2 σ 1 )y 2 +ξ (-qκ 1 κ 2 σ 3 + c 3 (q 2 κ 1 θ 1 + κ 2 θ 2 ))y + 2qθ 1 θ 2 +q(qc 3 κ 1 y -θ 2 )(y -c 1 )(y -c 2 ) = (ξ(y -c 4 ) - 1 κ 1 κ 2 c 3 (qc 3 κ 1 y -θ 2 )) × ×(κ 1 κ 2 ξ(y -c 3 )(c 3 yκ 2 -qθ 1 ) -κ 1 κ 2 c 3 (y -c 1 )(y -c 2 ))
One checks then that one of these factors divides the numerator of qκ 1 ỹ2 + ãỹ + qθ 1 . This yields the simpler expression

(4.12) ξ = c 1 c 2 κ 1 θ 1 ξ (ξ(y -qθ1 c1κ2 ) -q κ2 (y -c 2 ))(ξ(y -qθ1 c2κ2 ) -q κ2 (y -c 1 )) (ξ(y -c 4 ) -q κ2 (y -θ2 qc3κ1 ))(ξ(y -c 3 ) -q κ2 (y -θ2 qc4κ1 ))
.

Proposition 4.3. Formulas (4.10) and (4.12) express φ as a birational transformation φ : (y, ξ) → (ỹ, ξ).

4.4.

Blowing up the spaces. The map φ, whose expression as a birational transformation we obtained in the preceding section, can be extended to a regular map between two surfaces, by blowing up the spaces X κ1,κ2,θ1,θ2 and X qκ1,κ2,qθ1,θ2 . For this we first extend the map to the set of endomorphisms such that E 2 = F 1 . As noted before, this set corresponds to the blown up point with coordinates (0, c 1 c 2 /θ 2 ). Take a matrix in a neighbourhood of E 2 = F 1 in the form

κ 1 z 2 + āz + θ 2 z -y z(γz + β) κ 2 z 2 + bz + θ 1
with y small and ξ -c

1 c 2 κ 1 /θ 2 = λy = (y -c 1 )(y -c 2 )/z 1 , or ā = -κ 1 y + κ 1 y -κ 1 (c 1 + c 2 ) -λθ 2 λy + c 1 c 2 κ 1 /θ 2
We can conjugate the matrix to put it in the form

κ 1 z 2 + az + θ 1 z -y z(γz + β) κ 2 z 2 + bz + θ 2 with a = ā + θ 2 -θ 1 y b = b + θ 1 -θ 2 y γ = σ 1 + κ 1 b + κ 2 ā + (κ 1 -κ 2 )(θ 1 -θ 2 ) y β = - σ 3 + θ 1 b + θ 2 ā y - (θ 1 -θ 2 ) 2 y 2 then putting ∆ = (qθ 1 -θ 2 )(qκ 1 -κ 2 ) + qβ ỹ = ∆ (qκ 1 -κ 2 )(-qa -y(qκ 1 -κ 2 )) + (-qγ + (qκ 1 -κ 2 )b) we obtain ã = (qκ 1 -κ 2 )(qθ 1 -θ 2 )qa -qyβ(qκ 1 -κ 2 ) + qγ(qθ 1 -θ 2 ) + qβb ∆ We keep only the main terms as y → 0 to get ∆ ∼ -q (θ1-θ2) 2 y 2 ỹ ∼ -q (θ1-θ2) (q 2 κ1-κ2)y ã ∼ θ1-θ2 y therefore κ 1 z1 ∼ qκ 1 ỹ2 + ỹã ∼ κ 2 ỹ (θ 1 -θ 2 ) (q 2 κ 1 -κ 2 )y ∼ κ 2 ỹ2 /q or ξ = q/κ 2
Thus we see that the map can be extended if we blow up the final space at (∞, q/κ 2 ). A similar computation shows that the inverse map also extends to the curve Ẽ2 = F1 if we blow up the initial space at the point (∞, q/κ 2 ).

Proposition 4.4. Let Y κ1,κ2,θ1,θ2 denote P 1 × P 1 blown up at the eight points

(c 1 , 0), (c 2 , 0), (c 3 , ∞), (c 4 , ∞) (0, c 1 c 2 /θ 1 ), (0, c 1 c 2 /θ 2 ), (∞, 1/κ 1 ), (∞, q/κ 2 )
minus the strict transforms of the four lines y = 0, ∞, ξ = 0, ∞. then the map φ extends to a regular isomorphism between the spaces Y κ1,κ2,θ1,θ2 and Y qκ1,κ2,qθ1,θ2 .

(c 1 , 0) (c 2 , 0) (c 3 , ∞) (c 4 , ∞) (0, c 1 c 2 /θ 1 ) (0, c 1 c 2 /θ 2 ) (∞, 1/κ 1 ) (∞, q/κ 2 ) y = 0 ξ = 0 y = ∞ ξ = ∞ Fig. 3. The space Y κ1κ2,θ1θ2
5. Discrete Painlevé equations 5.1. Sakai's theory. We recall here the basic facts about the relation between surface theory and discrete Painlevé equations, according to Sakai [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]. One starts with a complex surface X, obtained by blowing up P 2 (C) at nine points x 1 , . . . , x 9 , thus X comes with a morphism π : X → P 2 (C). Degenerate configurations, where some of the points x i are infinitely close, are allowed. The Picard group of such a surface is a free abelian group of rank 10, generated by E 0 = π -1 (L), the inverse image of a generic line in P 2 (C), and E i = π -1 (x i ), i = 1, . . . , 9. The intersection form on Pic(X) satisfies

E 0 .E 0 = 1, E i .E i = -1, i = 1, . . . , 9, E i .E j = 0, i = j
The canonical divisor class is

K X = -δ = -3E 0 + E 1 + . . . + E 9
and its orthogonal δ ⊥ in Pic(X) is a rank 9 lattice, generated by

E i -E i+1 , i = 1, . . . , 8, E 0 -E 1 -E 2 -E 3
which form the simple roots in an affine root system of type E inside Pic(X)

E 1 -E 2 E 2 -E 3 E 3 -E 4 E 4 -E 5 E 5 -E 6 E 6 -E 7 E 7 -E 8 E 8 -E 9 d d d d d d d d d E 0 -E 1 -E 2 -E 3
The surface X is called a generalized Halphen surface by Sakai, if the linear system |-K X | has a unique divisor, whose irreducible components are orthogonal to K X . Let D i , i ∈ I be these components, then they generate an affine subroot system R in δ ⊥ ⊂ Pic(X), such that its orthogonal is also an affine subroot system R ⊥ ⊂ δ ⊥ . These surfaces are classified according to their corresponding root systems. Let X and X be two such surfaces, corresponding to the same root system R, where the irreducible components of the canonical divisor are labelled D i , i ∈ I, resp. Di , i ∈ I, then Sakai's main result is that (up to some little twists, see [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], Theorem 25 for a precise statement), every isometry of the Picard groups φ : Pic(X) → Pic( X) which preserves the components of the anticanonical divisor, φ((D i , i ∈ I)) = ( Di , i ∈ I) comes from a unique isomorphism of surfaces φ : X → X. Such isometries are given by elements of the extended Weyl group of the dual root system R ⊥ (i.e. the group generated by the Weyl group of R ⊥ and the automorphisms of the Dynkin diagram). When viewed at the level of P 2 (C), the isomorphisms φ : X → X give rise to birational transformations P 2 (C) → P 2 (C). The discrete Painlevé equations correspond to the translations in the affine Weyl group W (R ⊥ ). On the other hand, transformations in the extended Weyl group of R give Bäcklund transformations. As an example, Sakai treats the case of the birational tranformation

ρ : (f, g) → ( f , ḡ) determined by ḡ = b 3 b 4 g (f -b 5 )(f -b 6 ) (f -b 7 )(f -b 8 ) f = b 7 b 8 f (ḡ -qb 1 )(ḡ -qb 2 ) (ḡ -b 3 )(ḡ -b 4 )
where b 1 , . . . , b 8 are some complex parameters and

q = b 3 b 4 b 5 b 6 b 1 b 2 b 7 b 8 .
He shows that this transformation corresponds to an isomorphism between A

(1) 3

surfaces, which arises from a translation in an affine Weyl group D

5 . Here A

(1) 3

and D

(1) 5

are two orthogonal affine root systems in E

8 . Remark that this transformation is actually considered as a birational tranformation of P 1 (C) × P 1 (C). Since P 2 (C) blown up at 9 points is isomorphic to P 1 (C) × P 1 (C) blown up at 8 points the two pictures are equivalent, but it is sometimes easier to do formal reasoning on P 2 (C) and to perform actual computations in P 1 (C) × P 1 (C).

As explained in [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF], the transformation ρ can be considered as a step in a nonlinear difference equation

(f n+1 , g n+1 ) = ρ(f n , g n )
and this difference equation leads, by taking a suitable continuous limit, to a rational differential system df /dx = p(x, f, g) dg/dx = q(x, f, g) which is equivalent to a Hamiltonian version of the Painlevé VI equation. The degenerescence scheme of discrete Painlevé equations corresponds to the inclusion of the root systems in E surfaces. We will now identify the map φ between Y κ1,κ2,θ1,θ2 and Y qκ1,κ2,qθ1,θ2 we have constructed as one of the discrete Painlevé maps. For this we will use intersection theory for surfaces obtained as blow-ups. Denote by F i , i = 1, . . . , 8 the divisors which are inverse images, respectively, of the points (c 1 , 0), (c 2 , 0), (c 3 , ∞), (c 4 , ∞), (0, c 1 c 2 /θ 1 ), (0, c 1 c 2 /θ 2 ), (∞, 1/κ 1 ), (∞, q/κ 2 ) in the blow ups. Similarly we denote by Fi , i = 1, . . . , 8 the ones in the space Y qκ1,κ2,qθ1,θ2 .

It will be convenient to identify, as in Sakai [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF], the surfaces with P 2 blown up at 9 points, called x i . This identification is as follows. Consider P 1 × P 1 with coordinates denoted by (f, g), blown up at the eight points We make the explicit identification (5.1)

c 1 = b 1 c 2 = b 2 c 3 = b 3 c 4 = b 4 c 1 c 2 /θ 1 = b 5 c 1 c 2 /θ 2 = b 6 1/κ 1 = b 7 q/κ 2 = b 8 f = ξ g = y
between our space and Sakai's. Consider also P 2 blown up at the eight points

x 1 = (a 1 : 0 : 1), x 2 = (a 2 : 0 : 1), x 3 = (a 3 : 0 : 1), x 4 = (0 : 1 : a 4 ), x 5 = (0 : 1 : a 5 ), x 6 = (1 : a 6 : 0), x 7 = (1 : a 7 : 0), x 8 = (0 : 1 : 0) further blown up on π -1 (0 : 1 : 0) at

x 9 = x z , z x = (0, a 8 )
The two spaces are identified according to the transformation

(5.2) b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 ; f, g = 1/a 2 1/a 3 1/a 1 a 8 a 6 a 7 -1/a 4 a 1 -1/a 5 a 1 ; y x -a 1 z , z x
See pages 173 and 215 in [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]. The correspondance between divisors is

E 2 = F 1 , E 3 = F 2 , E 4 = F 5 , E 5 = F 6 , E 6 = F 7 , E 7 = F 8 , E 9 = F 4 , E 0 -E 1 -E 8 = F 3
D 0 = E 8 -E 9 D 1 = E 0 -E 6 -E 7 -E 8 D 2 = E 0 -E 1 -E 2 -E 3 D 3 = E 0 -E 4 -E 5 -E 8
which form a root system of type A The orthogonal is a root system of type D

5 , with generators (5.4) We denote by Ẽ and F the objects corresponding to the target space of the map φ.

α 0 = E 0 -E 1 -E 8 -E 9 , α 1 = E 2 -E 3 , α 2 = E 1 -E 2 α 3 = E 0 -E 1 -E 4 -E 6 , α 4 = E 6 -E 7 , α 5
We saw that φ maps F 6 to F8 and F 7 to F5 . We will compute the matrix of the map induced by φ between the Picard groups.

5.3.

Let us start by computing the image of the curve F 1 = E 2 . This curve consists of matrices of the form

A(z) = κ 1 z 2 + az + θ w(z -c 1 ) w -1 z(γz + β) κ 2 (z -c 1 )(z -θ 2 /c 1 κ 2 )
where a is a local parameter along the curve and w an irrelevant constant which can be removed by conjugation with a diagonal matrix but is convenient for computations (this is the "gauge freedom" in [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF]). The parameters γ and β are affine functions of a. Recall that

∆ = (qκ 1 -κ 2 )(qθ 1 -θ 2 ) + qβ then ỹ = ∆ qa(qκ 1 -κ 2 ) -(qκ 1 -κ 2 )(-c 1 -θ 1 /a 1 κ 1 ) + c 1 (qκ 1 -κ 2 ) 2 + γ(qκ 1 -κ 2 )
The critical value to be investigated are ỹ = 0, c i , ∞. The case ỹ = 0 corresponds to Ẽ1 = F2 so there is an intersection with F6 = E 5 . In the case ỹ = c 1 , the matrix

A(c 1 ) has the form S 0 T 0 , therefore Ã(c 1 ) = (qθ 1 -θ 2 )W -qW (qθ 1 -θ 2 )Z -qZ
with W = (qκ 1 -κ 2 )qS + βT , thus ỹ = c 1 if W = 0, and then z1 = 0, therefore the image does not intersect F1 = Ẽ2 . In the case y = c i , with i = 1, 2, 3, the matrix A(c i ) has rank 1 so

A(c i ) = rG G rH H
for some constants H and G (independent of a), and

Ã(c i ) = ((qθ 1 -θ 2 )r -β)((qκ 1 -κ 2 )G -qH) (qr + qκ 1 -κ 2 )((qκ 1 -κ 2 )G -qH) (qθ 1 -θ 2 )r -β)(βG + qθ 1 -θ 2 ) (qr + qκ 1 -κ 2 )(βG + qθ 1 -θ 2 )
The 12 coefficient vanishes only if qr + qκ 1 -κ 2 = 0, and then z2 = 0, therefore the image intersects Fi if i = 2 but not i = 3, 4, and intersects Ẽ1 . Finally for some value of a the equation ỹ = ∞ holds and the image intersects F7 . We conclude from these considerations that, in the Picard groups of the surfaces,

φ(E 2 ) = 2 Ẽ0 -Ẽ1 -Ẽ3 -Ẽ5 -Ẽ6 -Ẽ8 .
Similar tedious computations, which we omit, lead finally to the matrix induced by the map φ between the Picard groups, which is, from the basis (E i ) i=0,...,9 to ( Ẽi ) i=0,...,9 given by

                6 2 2 2 3 0 0 3 2 1 -2 0 -1 -1 -1 0 0 -1 -1 0 -2 -1 0 -1 -1 0 0 -1 -1 0 -2 -1 -1 0 -1 0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 -3 -1 -1 -1 -2 0 0 -1 -1 -1 -3 -1 -1 -1 -1 0 0 -2 -1 -1 0 0 0 0 0 1 0 0 0 0 -2 -1 -1 -1 -1 0 0 -1 0 0 -1 0 0 0 -1 0 0 -1 0 0                
One checks that this transformation preserves the components D i (cf (5.3))

φ(D 0 ) = D2 φ(D 2 ) = D0 φ(D 1 ) = D3 φ(D 3 ) = D1
while its action on the orthogonal root system (5.4) is a translation in the affine Weyl group, namely

φ(α 0 ) = α 0 , φ(α 1 ) = α 1 , φ(α 2 ) = α 2 , φ(α 3 ) = α 3 , φ(α 4 ) = α 4 -δ, φ(α 5 ) = α 5 + δ with δ = α 0 + α 1 + 2α 2 + 2α 3 + α 4 + α 5
This translation has the following reduced decomposition, where the w i are the simple reflexions associated with the roots α i , and σ is the automorphism of the Dynkin diagram such that σ(α 0 ) = α 1 , σ(α 4 ) = α 5 .

(5.5) φ = σw 4 w 3 w 2 w 0 w 1 w 2 w 3 w 4

We summarize the results of this section in the following.

Proposition 5.1. The transformation φ induces, on the Picard group of the surfaces, a translation whose reduced decomposition is given by φ = σw 4 w 3 w 2 w 0 w 1 w 2 w 3 w 4 Remark 5.2. The discrete Painlevé equation of [START_REF] Jimbo | A q-analog of the sixth Painlevé equation[END_REF] corresponds to the translation ψ given by

ψ(α i ) = α i , i = 0, 1, 4, 5 ψ(α 2 ) = α 2 -δ, ψ(α 3 ) = α 3 + δ
The translations φ and ψ are not conjugate in the Weyl group.

5.4.

As a check, we will obtain formulas (4.10), (4.12) using (5.5) and the explicit expressions for the birational transformations corresponding to the reflexions. After Sakai, we have the following formulas for the transformations corresponding to the roots w 0 : a 8 a 1 a 2 a 3 a 4 a 5 a 6 a 7 ; x : y : z → 1/a 1 1/a 8 a 2 a 3 a 1 a 4 a 5 a 1 a 6 a 8 a 7 a 8

; x(x -a 1 z) : y(x -z/a 8 ) : z(x -a 1 z) w 1 , w 4 , w 5 are the transpositions of parameters ; f, b5 b7 g f -b7

f -b5

The automorphism σ is the inversion This gives (4.10), (4.12), using the correspondance (5.1) and after simplifying the expression for ḡ, noting that f -b 5 divides the denominator, while f -b 8 divide the numerator.

5.5.

Finally the results of the preceding section allow one to derive relations for the Verblunsky coefficients. Indeed one has det A n (z) = q n V (z)W (z) thus one can take c 1 = ā/q, c 2 = 1/b c 3 = b/q, c 4 = 1/b. From (3.21) one sees that the coordinate y corresponds to αn(ā-bq n ) αn+1(a-bq n+1 ) . According to equations (3.22), (3.24), one finds that κ 1 = bq n+1 , κ 2 = aq, θ 1 = bq n , θ 2 = ā. We conclude that these expressions satisfy a discrete Painlevé equation.

Recovering the differential equation

We shall now explain how to obtain a differential system as a limit of the discrete dynamical system given by (4.10), (4.12). For this we introduce a complex variable t, and consider the parameters (tκ 1 , κ 2 , tθ 1 , θ 2 , c 1 , c 2 , c 3 , c 4 ). Consider the map φ of Proposition (4.3) as a birational map (y(tq), ξ(tq)) = φ(y(t), ξ(t)). Using the parametrisation q = 1 -,

qκ 1 = t(1 + K 1 ), κ 2 = 1 + K 2 qθ 1 = t(1 + Θ 1 ), θ 2 = 1 + Θ 2 , c i = 1 + C i , i = 1, . . . , 4 y = 1 + u, ξ = v
Letting go to zero gives a differential system in the variable t. We omit the details of the lengthy computation, and give the final result:

t(1 -t) du dt = tv(u -C 3 )(u -C 4 ) -t(t -1)v -1 (u -C 1 )(u -C 2 ) t(1 -t) dv dt = v [2u + 2t(K 2 -Θ 1 ) + C 1 + C 2 -K 1 -Θ 1 ] + 2u(t -1) + C 2 + C 1 -t(C 3 + C 4 ) + v -1 [2ut -t(1 + K 2 ) + 2(1 + Θ 2 ) -C 3 -C 4 ]
Finally, we come back to the original problem about the Fourier transform and the scattering problem for phase shifts which are quotients of Gamma functions, like (1.2). First note that in the case of the weight w given by (3.3), one has, from In order to recover (1.2) as a limit of the phase shift associated with the weight w we make the substitution q → 1 -, a → 1 + a, b → 1 + b where a and b are real numbers, before letting → 0. The final system is then

Ff (x) = 2 ∞ 0 2 ∞ 0 √

 2020 cos(2πxy)f (y)dy.One can rewrite this as an isometry of the space L 2 (]0, +∞[, dx/x), by putting f (x) = g(x)/ √ x, to obtain Fg(x) = xy cos(2πxy)g(y)dy/y

2π 0 e

 0 -ikθ dµ(θ) = c k and φ n (z) the sequence of orthogonal polynomials obtained by the Gram-Schmidt procedure from 1, z, z 2 , . . ., normalized by φ n (z) = z n + . . . This normalization will be more convenient for our purposes than the usual one, where the polynomials have L 2 -norm 1. Denote σ n = |φ n | 2 dµ and α n = φ n (0).

Fig. 1

 1 Fig. 1 Dynkin diagram of the affine root system E (1) 8

( 1 ) 8 .

 18 The degenerescence to Painlevé differential equations is also discussed by Sakai.

5. 2 .

 2 According to the classification of Sakai, the spaces Y κ1,κ2,θ1,θ2 are examples of A(1) 3

( 0 ,

 0 b 1 ), (0, b 2 ), (∞, b 3 ), (∞, b 4 ), (b 5 , 0), (b 6 , 0), (b 7 , ∞), (b 8 , ∞)

E 1

 1 is the line y = a 3 , and E 8 -E 9 the line ξ = ∞. The components of the anticanonical cycle are(5.3)

= E 4 -E 5 forming

 45

w 1 : 2 : b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 ;b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 ; f, g → b 1 b 2 b5 b7 b 3 b5 b7 b 4 b 7 b 6 b 5 b 8

 1288238 (a 2 , a 3 ); w 4 : (a 4 , a 5 ), w 5 : (a 6 , a 7 ) Passing to P 1 × P 1 with (5.2), w 2 and w 3 are w

b 1 b 2 b 3 b 4 b 5 b 6 b 8 b 7 ; f, g → b 1 b 2 b 3 b 4 b 5 b 6 b 8 b 7 ; b 7 b 8 = b 1 b 2 b 8 b 5 f

 7785 /f, b 3 b 4 /gWe can now compute φ induced by the translation (5.5). One getsf (f (g -b1b8 b5 ) -b 8 (g -b 1 ))(f (g -b2b8 b5 ) -b 8 (g -b 2 ) (f (g -b 3 ) -b 8 (g -b3b5 b8 )(f (g -b 4 ) -b 8 (g -

( 3 .

 3 [START_REF] Van Assche | Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials[END_REF],(3.24), and (3.4),(3.5),κ 1 = bq, θ 1 = b, κ 2 = aq, θ 2 = ā c 1 = ā/q, c 2 = b/q, c 3 = 1/b, c 4 = 1/a

t( 1

 1 -t) du dt = tv(u + b)(u + a) -t(t -1)v -1 (u -a + 1)(u -b + 1) t(1 -t) dv dt = v [2u + 2t(a -b -1) + a -b + 3] + 2u(t -1) + a + b + 2 + t(a + b) + v -1 [(2u -a)t + 2 + 3a + b]