Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulae for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.


Introduction and results
Throughout this paper we denote H N (I) the set of hermitian matrices of size N = 1, 2, . . . with eigenvalues in the interval I ⊆ R. In particular H N (I) can be endowed with the Lebesgue measure (1.1) The Jacobi Unitary Ensemble (JUE) is defined by the following measure on H N (0, 1) with parameters α, β satifying Re α, Re β > −1. The normalizing constant Γ(α + k + 1)Γ(β + k + 1) Γ(α + β + 2N − k) (1.3) ensures that dm J N has total mass 1; the above integral can be computed by a standard formula [18] in terms of the norming constants h J ℓ of the monic Jacobi polynomials, see (4.2). If α and β are integers, so that M α = α + N and M β = β + N are integers, the probability measure (1.2) describes the distribution of the matrix X = (W A + W B ) −1/2 W A (W A + W B ) −1/2 ∈ H N (0, 1) where W A = A † A and W B = B † B are the Wishart matrices associated with the random matrices A, B of size M α × N, M β × N respectively, with i.i.d. Gaussian entries [29].
Remark 1.1. Although (1.4) is defined only for Re α ± ℓ i=1 k i > −1, Re β > −1, it will be clear from the formulae of Corollary 1.6 below that the JUE correlators extend to rational functions of N, α, β.

JUE correlators and Hurwitz numbers
Our first result gives a combinatorial interpretation for the large N topological expansion [24,25,27] of JUE correlators (1.4). This provides an analogue of the classical result of Bessis, Itzykson and Zuber [13] expressing correlators of the Gaussian Unitary Ensemble as a generating function counting ribbon graphs weighted according to their genus, see also [24]. At the same time, it is more similar in spirit (and actually a generalization, see Remark 1.10) of the analogous result for the Laguerre Unitary Ensemble, whose correlators are expressed in terms of double monotone Hurwitz numbers [16], and (for a specific value of the parameter) in terms of Hodge integrals [21,19,30]; in particular in [30] we provide an ELSV-type formula [23] for weighted double monotone Hurwitz numbers in terms of Hodge integrals.
Our description of the JUE correlators involves triple monotone Hurwitz numbers, which we promptly define; to this end let us recall that a partition is a sequence λ = (λ 1 , . . . , λ ℓ ) of integers λ 1 ≥ · · · ≥ λ ℓ > 0, termed parts of λ; the number ℓ is called length of the partition, denoted in general ℓ(λ), and the number |λ| = ℓ j=1 λ j is called weight of the partition. We shall use the notation λ ⊢ n to indicate that λ is a partition of n, i.e. |λ| = n.
We denote S n the group of permutations of {1, . . . , n}; for any λ ⊢ n let cyc(λ) ⊂ S n the conjugacy class of permutations of cycle-type λ. It is worth recalling that the centralizer of any permutation in cyc(λ) has order where the symbol | · | denotes the cardinality of the set. Hurwitz numbers were introduced by Hurwitz to count the number of non-equivalent branched coverings of the Riemann sphere with a given set of branch points and branch profile [36]. This problem is essentially equivalent to count factorizations in the symmetric groups with permutations of assigned cycle-type and, possibly, other constraints. It is a problem of long-standing interest in combinatorics, geometry, and physics [44,31,35,34,32,33,3].
The type of Hurwitz numbers relevant to our study is defined as follows.
The proof is in Section 2. There is a similar result for the Laguerre Unitary Ensemble (LUE) [16] which is recovered by the limit explained in Remark 1.10. However, the proof presented in this paper uses substantially different methods than those employed in [16]; in particular our proof is completely self-contained and uses the notion of multiparametric weighted Hurwitz numbers, see e.g. [35] and Section 2.1.

Computing correlators of hermitian models
To provide an effective computation of the JUE correlators we first consider the general case of a measure on H N (I) of the form is a smooth function of x ∈ I • (the interior of I) and we assume that exp V (x) = O |x − x 0 | −1+ε for some ε > 0 as x ∈ I • approaches a finite endpoint x 0 of I; further, if I extends to ±∞ we assume that V (x) → −∞ fast enough as x → ±∞ in order for the measure (1.9) to have finite moments of all orders, so that the associated orthogonal polynomials exist. The expression tr V (X) in (1.9) for an hermitian matrix X is defined via the spectral theorem. The JUE is recovered for I = [0, 1] and V (x) = α log x + β log(1 − x), Re α, Re β > −1.
Introduce the cumulant functions which are analytic functions of z 1 , . . . , z ℓ ∈ C \ I, symmetric in the variables z 1 , . . . , z ℓ . To simplify the analysis it is convenient to introduce the connected cumulant functions from which the cumulant functions can be recovered by We now express the connected cumulant functions in terms of the monic orthogonal polynomials P ℓ (z) = z ℓ + . . . uniquely defined by and of the 2 × 2 matrix which is the well-known solution to the Riemann-Hilbert problem of orthogonal polynomials [28]; it is an analytic function of z ∈ C \ I. 16) with Y N (z) as in (1.15). Then the connected cumulant functions (1.11) are given by where prime in the second formula denotes derivative with respect to z and cyc((ℓ)) in the last formula is the set of ℓ-cycles in the symmetric group S ℓ .
The proof is given in Section 3. Formulae of this sort for correlators of hermitian matrix models have been recently discussed in the literature, see e.g. [26,20]. They are directly related to the theory of tau functions (formal [20] and isomonodromic [8,30]) and to topological recursion theory [14,27,5,4]. Incidentally, similar formulae also appear for matrix models with external source [41,10,6,7,11]. In Section 3 we provide a direct derivation based on the Riemann-Hilbert characterization of the matrix Y N (z).
We can apply these formulae to the Jacobi measure dm J N , see (1.2), for which the support is I = [0, 1]. Therefore we can expand the cumulants near the points z = 0 or z = ∞; the expansion at z = 1 could be considered but we omit it as it is recovered from the one at z = 0 by exchanging α, β, see (1.2). Using the definition in (1.10) and (1.11), we obtain the generating functions for the JUE connected correlators (1.4), namely On the other hand, performing the same expansion on the right hand side of the expressions for the cumulants in Theorem 1.5, we have an explicit tool to compute the correlators.
For the specific case of Jacobi polynomials, we prove in Section 4 (Proposition 4.1) that at z = ∞ the matrix R(z) has the Taylor expansion (valid for |z| > 1) of the form R(z) = T −1 R [∞] (z)T and the Poincaré asymptotic expansion R(z) ∼ T −1 R [0] (z)T at z = 0 valid in the sector 0 < arg z < 2π. Here T is the constant matrix with h J ℓ given in (4.2), and the series R [∞] (z), (1.23) Here 4 F 3 is the generalized hypergeometric function, and we use the rising factorial For example, the first few terms read . (1.28) Since the constant conjugation by T in (1.22) of the matrix R(z) does not affect the formulae of Theorem 1.5 (see also Section 4) we obtain the following corollary, which provides explicit formulae for the generating functions of the correlators.
The proof is in Section 4.2, and is obtained from the formulae of Theorem 1.5 by expansion at z i → ∞, 0. In this corollary, the formulae on the right hand side are interpreted as power series expansions at z = 0 or z = ∞. To this end, we remark that for ℓ ≥ 2 these series are well defined, as it follows from the fact that the corresponding analytic functions are holomorphic in (C \ I) ℓ and in particular regular along the diagonals z a = z b for a = b.
The coefficients of R [0] (z) and R [∞] (z) are rational functions of N, α, β and we conclude that JUE correlators extend to rational functions of N, α, β.
Examining more closely the formula for F 1,∞ we see that (1.25). Reasoning in the same way for F 1,0 (z) we obtain for more details see Proposition 4.4. Thus the formulae of Corollary 2.7 extend the connection between JUE moments tr X k and Wilson polynomials described in [17] to the JUE multi-point correlators tr X k 1 · · · tr X k ℓ .

Remark 1.8 (JUE mixed correlators).
We could consider more general generating functions as follows; take q, r, s ≥ 0 with q + r + s > 0 and expand the cumulant function C q+r+s (z 1 , . . . , z q , w 1 , . . . , w r , y 1 , . . . , y s ) (1.37) for the Jacobi measure as z i → ∞, w i → 0, y i → 1, to obtain the generating function It is then clear that we can compute the coefficients of such series in terms of the matrix series R [0] , R [∞] , and thus of Wilson polynomials, by the formulae of Theorem 1.5; note that the expansion of R(z) at z = 1 is obtained from R [0] by exchanging α with β and z with 1 − z.
Example 1.9. From the formulae of Corollary 1.6 we can compute .
With the substitution α = (c α − 1)N and β = (c β − 1)N we have the large N expansion Matching the coefficients as in Theorem 1.3 we get the values for h c g=0 (λ = (1, 1, 1), µ, ν) (the connected Hurwitz numbers defined in Remark 1.4) reported in the following table; For example, the numbers in the first row (µ = (3)) can be read from the following factorizations in Similarly we can compute from Corollary 1.6 and from Theorem 1.3 we recognize the connected Hurwitz numbers tabulated above.
Remark 1.10 (Laguerre limit). There is a scaling limit of the JUE correlators to the LUE correlators; if k 1 , · · · , k ℓ are arbitrary integers we have Therefore the results of the present work about the JUE directly imply analogous results for the LUE; these results are already known from [16,30]. See also Remark 2.9.

JUE Correlators and Hurwitz Numbers
In this section we prove Theorem 1.3. For the proof we will consider the so-called multiparametric weighted Hurwitz numbers; this far-reaching generalization of classical Hurwitz numbers was introduced and related to tau functions of integrable systems in several works by Harnad, Orlov [35], and Guay-Paquet [34], after the impetus of the seminal work of Okounkov [44].

Multiparametric weighted Hurwitz numbers
Let C[S n ] be the group algebra of the symmetric group S n ; namely, C[S n ] consists of formal linear combinations with complex coefficients of permutations of {1, . . . , n}. We shall need two important type of elements of C[S n ], which we now introduce. For any λ ⊢ n denote where we recall that cyc(λ) ⊂ S n is the conjugacy class of permutations of cycle-type λ. It is well known [45] that the set of C λ for λ ⊢ n form a linear basis of the center Z(C[S n ]) of the group algebra. The second class of elements consists of the Young-Jucys-Murphy (YJM) elements [39,43] J a , for a = 1, . . . , n, defined as . . , n} switching a, b and fixing everything else. Although singularly the YJM elements are not central, they commute amongst themselves, and symmetric polynomials of n variables evaluated at J 1 , . . . , J n generate Z(C[S n ]). Indeed the following relation [39] With these preliminaries we are ready to introduce the class of multiparametric Hurwitz numbers [35,34,9] which we need. Fix the real parameters γ 1 , . . . , γ L and δ 1 , . . . , δ M (L, M ≥ 0) and collect them into the rational function . (2.4) Then, the (rationally weighted) multiparametric (single) Hurwitz numbers H d G (λ), associated to the function G in (2.4) and labeled by the integer d ≥ 1 and by the partition λ ⊢ n, are defined by

Generating functions of multiparametric Hurwitz numbers in the Schur basis
The following result (see [35]) expresses the generating functions of multiparametric weighted Hurwitz numbers in the Schur basis. In this context, the latter is regarded as the basis {s λ (t)} (λ running in the set of all partitions) of the space of weighted homogeneous polynomials in t = (t 1 , t 2 , . . . ), with deg t k = k, whose elements are where the complete homogeneous symmetric polynomials h k (t) are defined by the generating series 1 In the following we shall denote P the set of all partitions.
of multiparametric weighted Hurwitz numbers (2.5) associated to the rational function (2.4) is equivalently expressed as where s λ (t) are the Schur polynomials (2.6) and the coefficients are given explicitly by dim λ being the dimension of the irreducible representation of S |λ| associated with λ.
Before the proof we give a couple of remarks.
1. In (2.10) and below we use the notation For example, the diagram of the partition λ = (4, 2, 2, 1) ⊢ 9 is depicted below; 2. There exist several equivalent formulae for dim λ, including the well-known hook-length formula; for later convenience we recall the expression Proof of Proposition 2.1. We need a few preliminaries. First we recall that Z (C[S n ]) is a semi-simple commutative algebra; a basis of idempotents is given by (see e.g. [45]) where χ µ λ are the characters of the symmetric group and C µ are given in (2.1). Namely (2.14) For any symmetric polynomial p(y 1 , . . . , y n ) in n variables, p(J 1 , . . . , J n ) belongs to Z (C[S n ]), as we have already mentioned; central elements are diagonal on the basis of idempotents and it is proven in [39] that where in the right hand side we denote p {j − i} (i,j)∈λ the evaluation of the symmetric polynomial p at the n values of j − i for (i, j) ∈ Z 2 in the diagram of λ ⊢ n; in the example λ = (4, 2, 2, 1) ⊢ 9 above, see (2.11), this denotes the evaluation p(0, 1, 2, 3, −1, 0, −2, −1, −3). We are ready for the proof proper. First note that by (2.15) and (2.10) we have Multiplying this identity by ℓ(µ) i=1 t µ i and summing over all partitions µ, on the left we obtain (2.8) and on the right, thanks to the well-known identity [42] we obtain (2.9). The proof is complete.
Remark 2.2. This result is used by the authors of [35] to prove that the generating function τ G (ǫ; t) is a one-parameter family in ǫ of Kadomtsev-Petviashvili tau functions in the times t; a tau function such that the coefficients of the Schur expansion have the form (2.10) is termed hypergeometric tau function. It is also worth remarking that the theorem stated here is a reduction of a more general result, proved in [34], dealing with generating functions of double (weighted) Hurwitz numbers. In this general setting, the corresponding integrable hierarchy is the 2D Toda hierarchy.

JUE partition functions
Let us introduce the formal generating functions of JUE correlators; the sum in the right hand side is a formal power series in u running over all partitions λ, with the combinatorial factor z λ defined in (1.5) 2 . We call Z + N (u) (resp. Z − N (u)) the positive (resp. negative) JUE partition function. Although it will not be needed in the following, we mention that these partition functions are Toda tau functions in the times u 1 , u 2 , . . . [1,2,18]. Our goal in this paragraph is to show that the JUE partition functions can be expressed in the form (2.9) for appropriate choice of G (see Corollary 2.7).
The first step is to expand the JUE partition functions in the Schur basis; this is achieved by the following well-known general lemma, whose proof we report for the reader's convenience. The idea of expanding a hermitian matrix model partition function over the Schur basis has been recently used in the computation of correlators in [38].
We first introduce the following notations for the Vandermonde determinant and for the characters of GL n ; again, we set λ i = 0 for all ℓ(λ) < i ≤ N .
where the Schur polynomials are defined in (2.6) and the coefficients are (2.25) Here x = (x 1 , . . . , x N ) and 26) where we use the standard decomposition dX = ∆ 2 (x)d N xdU of the Lebesgue measure into eigenvalues x = (x 1 , . . . , x N ) and eigenvectors U ∈ U N of the hermitian matrix X = U XU † , with dU a Haar measure on U N (whose normalization is irrelevant as it cancels in (2.26) between numerator and denominator). The proof follows by an application of the identity which is nothing but a form of Cauchy identity, see e.g. [46].

Remark 2.4. By applying Andréief identity
it is straightforward to show that the coefficients c λ,N in (2.25) can also be expressed as

29)
see also [38]. However, for our purposes it is more convenient to work with the representation (2.25).
Applying this general lemma to I = [0, 1] and V (x) = α log x + β log(1 − x) we can expand the positive and negative JUE partition functions in the Schur basis as For the negative coefficients c − λ,N we shall use the following elementary lemma.
Proof. The proof follows from the following chain of equalities; In the first step we have shuffled the columns as j → N −j +1, then we have multiplied both numerator and denominator by (x 1 · · · x N ) N +λ 1 , and finally we have applied the definition (2.23).
For the simplification of the coefficients (2.31) we rely on the following Schur-Selberg integral (2.34) for which we refer e.g. to [42, page 385]. The above allows us to prove the following proposition. Proposition 2.6. We have

(2.35)
Proof. We start with c + λ,N ; using (2.31), (2.34), and (2.12) we compute We remind that we are using the notation (1.27) for the rising factorial. For c − λ,N we first note that, thanks to Lemma 2.5 and (2.34), we have (2.37) then with similar computations as above we obtain This proposition enables us to identify the Jacobi generating function (2.21) with the generating function of multiparametric weighted Hurwitz numbers in (2.8) as follows.
Proof. We first note that we can rewrite the expansion (2.30) as with the sum over all partitions P and no longer restricted to ℓ(λ) ≤ N ; this is clear as c ± N,λ = 0 whenever N = 0, 1, 2, . . . and ℓ(λ) > N . Then the proof is immediate by the formula (2.10) for the coefficients r (G,ǫ) λ , since (2.35) can be rewritten as . (2.43)

Hurwitz numbers h g (λ, µ, ν) and multiparametric Hurwitz numbers
We now connect the multiparametric Hurwitz numbers (2.5) for the functions G ± (z), appearing in Corollary 2.7, with the counting problem in Definition 1.2.
Proof. We apply (2.3) to the first two factors of the following to get (1 + ǫJ a )(1 + ǫγJ a ) (2.45) By definition (2.5), extracting the coefficient of ǫ d C λ and dividing by z λ we obtain H d G (λ); therefore where d, r, g in this identity are related via The proof is complete by the identity z λ |cyc(λ)| = n!.
Remark 2.9. Let us note that letting c β → ∞ in the functions G ± of Corollary 2.7 we have G + (z) → (1 + z)(1 + z/c α ) and G − (z) → (1 + z)/(1 − z/(c α − 1)). The Hurwitz numbers corresponding to these limit functions can be identified as in Proposition 2.8 in terms of double strictly (+) or weakly (−) Hurwitz numbers, respectively. Thus, bearing in mind the scaling limit for β → ∞ of JUE correlators to the correlators of the Laguerre Unitary Ensemble of Remark 1.48, the Theorem 1.3 recovers the results of [16].

Computing correlators of Hermitian models
In this section we prove 3 Theorem 1.5. The strategy is based on the observation that setting .

(3.2)
Here and below it is assumed that z i ∈ I.

Orthogonal polynomials on the real line and unitary-invariant ensembles
We denote P ℓ (z) the monic orthogonal polynomials, h ℓ = I P 2 ℓ (x)e V (x) dx, see (1.14), and their Cauchy transforms. The matrix , (3.4) introduced in (1.15), is an analytic function of z ∈ C \ I. It satisfies the jump condition where we use the notation

6)
and I • is the interior of the interval I. As z → ∞ we have where we denote 1 = 1 0 0 1 and σ 3 = 1 0 0 −1 . Lastly, we recall the Christoffel-Darboux identity expressing the kernel (Christoffel-Darboux kernel ) of the orthogonal projector onto the space of polynomials of degree < N in the Hilbert space L 2 (I, e V (x) dx). Therefore, the Christoffel-Darboux kernel can be conveniently rewritten in terms of the matrix Y N (z) in (3.4) as which is independent of the choice of boundary value of Y N because of (3.5). Next, we need to recall the connection of orthogonal polynomials to the theory of unitary-invariant ensembles of random matrices. The main point which is relevant for our present purposes is that [18] where it is convenient to explicitly express the dependence of P ℓ = P V ℓ and h ℓ = h V ℓ on the potential V . Therefore, introducing the modified potential but the first term vanishes by orthogonality because P Vt,z i (x) are normalized to be monic and, therefore, is a polynomial of degree strictly less than i.

Case ℓ = 1
It follows from (3.9) that In the following we shall use the notation for the jump of a function f across I, namely f ± (x) := lim ǫ→0 + f (x ± iǫ). The next lemma is well known, see e.g. [15], and it is proven here for the reader's convenience.
Lemma 3.2. We have Proof. Let us denote ′ := ∂ x . It follows from the jump condition (3.5) for Y N that Therefore we compute The last term vanishes and so, by the cyclic property of the trace, we have which is easily seen to be equivalent, up to multiplying by −1/(2πi), to (3.15).
We are ready for the proof of the case ℓ = 1. In such case, t = t 1 , z = z 1 and V t,z (x) = V (x) + t/(z − x). By (3.12), Lemma 3.1, and (3.8), we have (3.21) where we denote explicitly the dependence of the Christoffel-Darboux kernel on the potential. Let Γ be an oriented contour in the complex plane which surrounds I in counterclockwise sense (i.e., I lies on the left of Γ) and leaves z outside (i.e., z lies to the right of Γ). Then, using Lemma 3.2 we get , (3.22) where Y N (·; t, z) is the matrix (3.4) for the potential V t,z . The last contour integral can be evaluated by a residue computation as It can be checked from (3.7) that the residue at x = ∞ vanishes. Therefore Evaluating this identity at t = 0, taking into account (3.2), we obtain exactly (1.17).

Case ℓ = 2
Let us first formulate a result that will be needed for all ℓ ≥ 2. Let Proof. Let us denote by Ω j (x; t, z) the left-hand side of (3.26). Using (3.5) we get the identities from which we readily ascertain that ∆Ω j (x; t, z) = 0 for all x ∈ R. Hence, Ω j (x; t, z) is a meromorphic function of x with a single simple pole at x = z j and which vanishes at x = ∞, because of (3.7), and so the statement follows.
Let us consider the case ℓ = 2, in which t = (t 1 , t 2 ), z = (z 1 , z 2 ), and V t,z (x) = V (x) + t 1 z 1 −x + t 2 z 2 −x . By the argument used for ℓ = 1, cf. (3.24), we obtain Next we have to take a derivative in t 2 : omitting the explicit dependence on t, z, we have (3.30) We use (3.26) to rewrite the first term inside the trace in the right-hand side as and the second term as where [A, B] := AB − BA is the commutator. The term in the last row exactly cancels with (3.31), and so, rearranging terms
Remark 3.4. We note here that since R(z) is a rank one matrix, the formulae of Theorem 1.5 for C c ℓ , ℓ ≥ 2, can be expressed in terms of the scalar quantities compare for instance with [22,48].

JUE correlators and Wilson Polynomials
In this section we prove Corollary 1.6. This is done by expanding the general formulae of Theorem 1.5 as z i → 0, ∞. To this end we consider the monic orthogonal polynomials for the Jacobi measure, which are the classical (monic) Jacobi polynomials satisfying the orthogonality property

Expansion of the matrix R
This paragraph is devoted to the proof of the following proposition.
Proposition 4.1. We have the Taylor expansion at z = ∞ where T is the constant matrix (1.22) and R [∞] (z) is the matrix-valued power series in z −1 in (1.23).
We have the Poincaré asymptotic expansion at z = 0 uniformly within the sector 0 < arg z < 2π where T is the constant matrix (1.22) and R [0] (z) is the matrix-valued (formal) power series in z in (1.23).
Looking back at the definition of the matrix R(z), we notice that it is sufficient to compute the expansions of the product of the Jacobi polynomials with their Cauchy transforms at the prescribed points. To this end, recall the explicit formula (4.1) for the monic Jacobi orthogonal polynomials, which can be rewritten as the Rodrigues' formula The Cauchy transforms P J ℓ (z) defined in (3.3) can be expanded as stated below. Lemma 4.2. The following relations hold true: where the first relation is a genuine Taylor expansion at z = ∞ whilst the second one is a Poincaré asymptotic expansion at z = 0 uniform in the sector 0 < arg z < 2π.
Proof. We start with the expansion (4.7) at z = ∞, which is computed as follows; In (i) we have expanded the geometric series and exchanged sum and integral by Fubini theorem, in (ii) we use that P J ℓ (z) is orthogonal to z j for j < ℓ, in (iii) we use the Rodrigues' formula (4.6), in (iv) we integrate by parts, in (v) we compute the derivative, and finally in (vi) we use the Euler beta integral. The computation at z = 0 is completely analogous, with the only difference that in (i) it is not legitimate to exchange sum and integral so this step holds only in the sense of a Poincaré asymptotic series. The next step is to compute the expansions of the products of the Jacobi polynomials and their Cauchy transforms. To this end it is convenient to study more in detail the properties of R(z). Proposition 4.3. The matrix Ψ N (z) := Y N (z)z ασ 3 /2 (1 − z) βσ 3 /2 satisfies the following linear differential equation and the matrix R(z) satisfies the following Lax differential equation, Here the matrix U (z) is explicitly given as ; therefore the Lax equation (4.11) follows from (4.10). The latter is a classical property of Jacobi orthogonal polynomials [37].
To prove Proposition 4.1 is equivalent to prove that R(z) ∼ R [p] (z) for p = ∞, 0 where R(z) = T R(z)T −1 . (4.15) It follows from the previous proposition that R(z) satisfies Introduce the matrices and write where we used that tr R(z) = 1, tr U (z) = 0. For the sake of brevity we omit the dependence on z in the sl 2 components. The Lax equation (4.16) yields the coupled first order linear ODEs Proof. The identification with the Wilson polynomials is obtained by comparing the recurrence relations (4.26) and (4.27) with the difference equation for this family of orthogonal polynomials, which reads where w(k) = W n (k 2 ; a, b, c, d) and The hypergeometric representation of A ℓ , B ℓ then directly follows from that of the Wilson polynomials in (1.36).
The above Proposition, together with the expansions (4.25), yields the first part of Proposition 4.1. The asymptotics of R(z) at z = 0 are obtained in a similar way. More precisely, we claim that the expansion at z = 0 of the entries of R(z) reads as From Theorem 1.5 we write the formula for C 1 (z) by using the differential equation (4.10) as where E 1,1 := 1 0 0 0 . In the last step we used the definition of R(z) = Y N (z)E 1,1 Y N (z) −1 , the cyclic property of the trace and the equation which follows from (4.10). Proof. We compute ∂ z [z(1 − z)tr (U (z)R(z))] = (1 − 2z)tr (U (z)R(z)) + z(1 − z)tr (U ′ (z)R(z)) + z(1 − z)tr (U (z)R ′ (z)). which can be checked directly from (4.12). The proof is complete.