Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlev\'e IV Equation

We consider a $q$-Painlev\'e IV equation which is the $A_4^{(1)}$-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ${}_2\varphi_1$ basic hypergeometric series and the other is given by ${}_2\psi_2$ bilateral basic hypergeometric series.


Introduction
In the 1900s, the Painlevé equations, P I , P II , . . . and P VI , were defined by P. Painlevé, R. Fuchs and B. Gambier. The Painlevé equations are nonlinear ordinary differential equations of second order that posses no movable singular point. There are many works that investigate the properties of the Painlevé equations [29,30,31,32,33] Discrete Painlevé systems are nonlinear ordinary difference equations of second order and known as discrete versions of Painlevé equations. It is known that there are three difference types (additive type, multiplicative type and elliptic type) for discrete Painlevé systems. There are also many works that investigate the properties of discrete Painlevé systems [8,16,17,37] In [37], H. Sakai introduced a geometric approach to the theory of Painlevé systems and showed the classifications of Painlevé equations and discrete Painlevé systems by the rational surfaces The rational surface can be identified with the space of initial condition, and the group of Cremona isometries associated with the surface generate the affine Weyl group.
Some discrete Painlevé systems have been found in the studies of random matrices [4,12,34] As one such example, let us consider the partition function of the Gaussian Unitary Ensemble of an n × n random matrix: where ∆(t 1 , · · · , t n ) is Vandermonde's determinant. Note that throughout this paper we assume n i=1 f (i) = 1, f (i) = 1, ∆(t 1 , · · · , t n ) = 1, (1.2) for an arbitrary function f (i) when n = 0. Here we choose we obtain the following difference equation [4,10,11,36]: (1.5) Equation (1.5) is referred to as a discrete Painlevé I equation, denoted by d-P I , and has the space of initial condition of type E (1) 6 . Such relations between discrete Painlevé systems and random matrices are well known. Now, we introduce a q-version of a partition function, using (1.1) as our reference. We consider ψ l,m n (l, n ∈ Z ≥0 , m ∈ Z, a ∈ C, c 1 ∈ R >0 ) given as The definitions of the q-definite integral ∞ −∞ d q t and the q-exponential function E q (t) appearing here are given at the end of this section. As in the case of (1.1), we can obtain a solution to a discrete Painlevé equation expressible in terms of ψ l,m n . Specifically, we have the following: (q-P IV ) [35,39] : (X n+1 X n − 1)(X n−1 X n − 1) = q −N +2n−m−1 a 0 a 1 3/2 a 2 2 (X n + q N −m a 1 1/2 )(X n + q −N +m a 1 −1/2 ) has the following solution: (1.8) Here The proof of Lemma 1.1 will be given in Appendix. Below, we investigate the solutions to d-P I and q-P IV from the viewpoint of orthogonal polynomials. First, however, we define orthogonal polynomials: Definition 1.1. A polynomial sequence (P n (t)) ∞ n=0 which satisfies the following conditions is called an orthogonal polynomial sequence over the field K, and each term P n (t) is called an orthogonal polynomial over the field K.
• deg(P n (t)) = n • There exists a linear functional L : K(t) → K which holds the orthogonal condition: where δ n,k is Kronecker's symbol. Here, h n is called the normalization factor and µ n = L[t n ] (n = 0, 1, . . . ) is called the moment sequence.
2. An orthogonal polynomial sequence whose coefficient of leading term is 1 is called a monic orthogonal polynomial sequence (MOPS). Let (P n (t)) ∞ n=0 be MOPS. P n (t) and its normalization factor h n are given as Here (µ n ) ∞ n=0 is the moment sequence and τ n is the Hankel determinant given as First, we reconsider the solution to d-P I appearing in (1.4). Let (P n (t)) ∞ n=0 be MOPS defined as The moment is given as 0 (n = 2k + 1), (1.14) and the normalization factor is Here D λ (z) is the parabolic cylinder function defined as which satisfies P n satisfies the following three-term recurrence relation: Substituting n = k in (1.13) and then applying partial integration on it, we obtain From compatibility conditions of (1.18) and (1.19), we find that is the solution to d-P I . Instead of (1.19) we can use a differential equation or a difference equation for P n (t). In any case, R n = h n /h n−1 essentially becomes a variable of the discrete Painlevé equation. We next reconsider the solution to q-P IV appearing in (1.8). The Hankel determinant expression of ψ l,m n is given by the following lemma: (1.21) Here, the entries are given as Proof . Equation (1.21) can be verified by the following calculation:  the solution to q-P IV can be rewritten as (1.29) Note that P 0,m n (t) defined as (1.27) is referred to as the discrete q-Hermite II polynomial (cf. [19]) and P 1,m n (t) is said to the kernel polynomial of P 0,m n (t) given by the Christoffel transformation. The definitions of the kernel polynomial and the Christoffel transformation will be given in the next section.
The solution to d-P I , (1.20), is given by the single orthogonal polynomial, while that to q-P IV , (1.29), is expressed by the two different orthogonal polynomials. From this viewpoint, the types of solutions to d-P I and q-P IV are different. In the past the solutions to discrete Painlevé systems expressed in terms of normalization factor of one type of orthogonal polynomial has been studied [1,2,3,4,12,34,38,41], but as far as I know, there is no study about one expressed in terms of normalization factors of two types of orthogonal polynomial. The purpose of this paper is to construct the method to give the solutions expressed in terms of normalization factors of two types of orthogonal polynomials. We note here that solutions to the Painlevé equations expressed in terms of normalization factors of two types of orthogonal polynomial is studied in [5,6]. This paper is organized as follows. In Section 2, we consider the compatibility conditions of an orthogonal polynomial and its kernel polynomial. In Section 3, we demonstrate with examples that from the compatibility condition given in Section 2 we can obtain the solution to the discrete Painlevé systems. Concluding remarks are given in Section 4.
Throughout this paper, we assume 0 < |q| < 1 and the expression "α is a constant" means dα/dt = 0, where t is the independent variable of the orthogonal polynomial. We use the following conventions of q-analysis [7,19] q-Shifted factorials: Jacobi theta function: q-Exponential function: Basic hypergeometric series: Bilateral basic hypergeometric series: Finally, we note that the following relations hold: The compatibility conditions associated with the Christoffel transformation In this section, we consider the compatibility conditions of an orthogonal polynomial and its kernel polynomial. Let (P n ) ∞ n=0 = (P n (t)) ∞ n=0 and (P n ) ∞ n=0 = (P n (t)) ∞ n=0 be MOPSs with linear functionals L andL over C given as respectively. µ n andμ n are the moments given as respectively. We refer to the transformation from P n toP n as the Christoffel transformation andP n as the kernel polynomial. On the other hand, we also refer to the transformation from P n to P n as the Geronimus transformation. First we consider the relations between P n andP n .
Lemma 2.1. The following relations hold: Proof . We first prove (2.5). The polynomial (t − c 0 )P n can be expressed with certain constants C j as and then it holdŝ where 0 ≤ k ≤ n. Therefore (2.5) holds. We next prove (2.6). Setting whereĈ j is a constant, we obtain where 0 ≤ k ≤ n − 1. Therefore (2.6) holds.
We define the constants α n , β n ,α n andβ n as tP n = P n+1 + α n P n + β n P n−1 , From (2.13) and (2.14), we obtain the following: From (2.17) and (2.18), we obtain When we give an orthogonal polynomial P n such that both α n and β n are rational functions of n (or q n ), we can regard (2.20) as a discrete Riccati equation. Similarly, when we give an orthogonal polynomialP n , (2.21) can be also regarded as a discrete Riccati equation. Therefore we find that the compatibility conditions of an orthogonal polynomial and its kernel polynomial can be related to the discrete Painlevé equation through the discrete Riccati equation. In the next section, we demonstrate this point with examples in the case where both α n and β n (or, α n andβ n ) are rational functions of n and in the case where both α n and β n (or,α n andβ n ) are rational functions of q n .

Relation between the compatibility conditions and discrete Painlevé systems
In this section, we show that from (2.20) and (2.21) we can obtain the solutions to discrete Painlevé systems. We demonstrate the construction by taking two examples. The first example is the Hermite polynomials in the case where both α n and β n (or,α n andβ n ) are rational functions of n. The second one is the discrete q-Hermite II polynomials in the case where both α n and β n (or,α n andβ n ) are rational functions of q n .
3.1 Example I: The case where (P n ) ∞ n=0 are the Hermite polynomials We define P n as where H n is the Hermite polynomial: The linear functionals and the moments are given as

4)
where D λ (z) is the parabolic cylinder function defined in (1.16). From the three-term recurrence relation: we obtain From (2.20), we obtain the following discrete Riccati equation: We consider the following difference equation [9,10,11,26,34,36]: (3.10) Equation (3.10) is referred to as a discrete Painlevé II equation, denoted by d-P II , and has the space of initial condition of type D 5 . d-P II admits a specialization to the discrete Riccati equation: (3.12) Therefore we obtain the following theorem: Theorem 3.1. d-P II (3.10) admits the following solution: X n = 2(n + 1) c 2 (c 1 + c 0 c 2 ) x n + 1, (n ∈ Z ≥0 ). (3.13) (3.14) Remark 3.1. The relation between d-P II (3.10) and MOPS is already known [34]. However in the previous paper d-P II relates to the MOPS P n (t) defined as |t|=1 P n (t)P k (1/t) e −(t+1/t)/a 2πit dt = h n δ n,k , (n ≥ k). We next consider the case whereP n is the Hermite polynomial: We assume here that c 0 is not a real number. The linear functionals are given as 18) and the moments are given by the following lemma: Lemma 3.1. The following equations hold:

21)
where D λ (z) is the parabolic cylinder function.
Proof . Equations (3.19) and (3.20) are obvious. We prove (3.21). Replacing t with √ 2c 2 −1 t+c 0 , we can rewrite µ 0 as Using the integral representation of the Legendre polynomial: where the contour C runs from −∞ to +∞ so that t = 0 lie to the right of the contour, we obtain Then the statement follows from From the three-term recurrence relation: we obtain From (2.21), we obtain the following discrete Riccati equation: Therefore we obtain the following theorem: Theorem 3.2. d-P II (3.10) admits the following solution: X n = 2(n + 1) c 2 (c 1 + c 0 c 2 ) y n + 1, (n ∈ Z ≥0 ).

Example III:
The case where (P n ) ∞ n=0 are the discrete q-Hermite II polynomials We define P n as P n (t) = h II n (c 1 t; q) c 1 n , (c 1 > 0), (3.31) where h II n is the discrete q-Hermite II polynomial: The linear functionals, the three-term recurrence relation and the discrete Riccati equation are given by respectively. For the moment sequences, the following lemma holds: Lemma 3.2. The following equations hold: Θ(−c 1 2 ; q 2 ) (q; q 2 ) k q k 2 c 1 2k (n = 2k), 0 (n = 2k + 1), Proof . Equation (3.38) is obvious. We prove (3.37). From we obtain we obtain Therefore we have completed the proof.
We obtain the following theorem: Theorem 3.3. q-P IV (1.7) admits the following solution: We find that the solution to q-P IV given in Theorem 3.3 coincides with one given in (1.29).

Example IV:
The case where (P n ) ∞ n=0 are the discrete q-Hermite II polynomials We consider the case whereP n is the discrete q-Hermite II polynomial: We assume that c 0 = q a for ∀ a ∈ Z. Then we have the linear functionals and the moments as (q; q 2 ) k q k 2 c 1 2k (n = 2k), 0 (n = 2k + 1). (3.51) Three-term recurrence relation and the discrete Riccati equation are given by Theorem 3.4. q-P IV (1.7) admits the following solution:

Concluding remarks
In this paper, we constructed the method to give the solutions to discrete Painlevé systems expressed in terms of normalization factors of two types of orthogonal polynomials and also presented some examples. It seems that the solutions of various discrete Painlevé systems can be constructed by using the method in this paper.
Before closing this paper, we briefly discuss the structure of the solutions of discrete Painlevé systems derived in this paper. It is well known that the τ functions play a crucial role in the theory of integrable systems including Painlevé systems [13,14,15,18,22,27,28,30,31,32,33]. Further, it is also known that the particular solutions to Painlevé systems are expressible in the form of ratio of determinants, and the determinants directly related with τ functions [20,21,23,25]. However we have not yet clarified the relation between the solutions obtained in this paper and the τ functions. For example, let us consider the q-P IV (1.7). In [40], T. Tsuda introduced the τ functions for q-Painlevé equations, including q-P IV (1.7), with affine Weyl group symmetry of type A (1) 4 . The solutions given in the following proposition are constructed by the method to construct the hypergeometric τ functions (cf. [20,21,23,25]). Therefore the determinants of this solution are directly related to the τ functions.

Proposition 4.1 ([24]
). When a 0 1/2 a 1 1/2 = q and N ≥ 0, q-P IV (1.7) has the following solutions: Here, F n,m is given as where A n,m and B n,m are periodic functions of period one for n and m, i.e., Comparing the configuration of ψ l,m n in (1.8) with one of φ n,m N in (4.1), we find that the relation between the determinants of the solutions given in this paper and the τ functions is not obvious. We note here that the solutions given in this paper are called molecule type solutions, whose determinant size depends on an independent variable and ones given in Proposition 4.1 are called lattice type solutions, whose determinant size does not depend on an independent variable. = I l,m q −1 I l+1,m+1 · · · q −(n−1) 2 I l+n−1,m+n−1 I l+1,m q −2 I l+2,m+1 · · · q −n(n−1) I l+n,m+n−1 . . . . . . . . . . . .

(A.2)
Here Proof . The statement follows from the following calculation: We next consider the linear relations of I n,m .
Lemma A.2. The following contiguity relations hold: Proof . Equation (A.13) follows immediately from (A.3). We next prove (A.14). Using the partial integration (1.42), we obtain Therefore (A.14) is derived as follows: respectively. We also introduce the following notation: It is easy to verify the following equalities from the definition: Note that (A.13) implies the following relation: Then we obtain the following lemma: X n+1 = 1 − q n+2 X n + ic 1 aq n−m+1 . (A.73) We note here that in the case of a 0 a 1 = q −2N , q-P IV (1.7) admits a specialization to the following discrete Riccati equation: This completes the proof of Lemma 1.1.