The Padé interpolation method applied to q-Painlevé equations II (differential grid version)

Abstract

Abstract. Recently we studied Padé interpolation problems of q-grid, related to q-Painlevé equations of type E (1) 7 , E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) . By solving those problems, we could derive evolution equations, scalar Lax pairs and determinant formulae of special solutions for the corresponding q-Painlevé equations. It is natural that the q-Painlevé equations were derived by the interpolation method of q-grid, but it may be interesting in terms of differential grid that the Padé interpolation method of differential grid (i.e. Padé approximation method) has been applied to the q-Painlevé equation of type D (1) 5 by Y. Ikawa. In this paper we continue the above study and apply the Padé approximation method to the q-Painlevé equations of type E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) . Moreover determinant formulae of the special solutions for q-Painlevé equation of type E (1) 6 are given in terms of the terminating q-Appell Lauricella function. IntroductionIn this paper we continue [13] and apply the Padé approximation method to the q-Painlevé equations of type E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) .1.1. The background of discrete Painlevé equations.In Sakai's theory [21] the discrete Painlevé equations were classified on the basis of rational surfaces connected to extended affine Weyl groups. There exist three types of discrete Painlevé equations in the classification: elliptic difference (e-), multiplicative difference (q-) and additive difference (d-). The discrete Painlevé equations of q-difference type are classified as follows:Here A → B means that B is obtained from A by degeneration. method to continuous Painlevé equations of type P VI , P V , P IV and Garnier system by using differential grid (i.e. Padé approximation).The Padé method for discrete Painlevé equations has been applied to the following types:( Remark 1. On the key points of the Padé methodThere are two key points to apply the Padé approximation/interpolation method [4,13,17,33]. The first key point is the appropriate choice of approximated/interpolated functions (see Table (2.2) and Remark 4). The second key point is to consider two linear q-difference three term relations (2.6) satisfied by the error terms of the Padé approximation/interpolation problems. Then the error terms can be expressed in terms of special solutions of q-Painlevé equations. Therefore the q-difference relations are the main subject in our study, and they naturally give the evolution equations, the Lax pairs and the special solutions for the corresponding q-Painlevé equations. Remark 2. On a connection between the Padé method and the theory of semiclassical orthogonal polynomialsThe connection between semiclassical orthogonal polynomials (classical orthogonal polynomials related to a suitable weight function) and Painlevé/Garnier systems has been demonstrated in [11]. It has been shown that coefficients of three term recurrence relations, satisfied by several semiclassical orthogonal polynomials, can be expressed in terms of sol...

Study of <i>q</i>-Garnier System by Padé Method

“…In this method, one can obtain the evolution equation, the Lax pair and some special solutions simultaneously, starting from a suitable Padé approximation (or interpolation) problem. This method has been applied [5,14,15,16,18,32] to various cases of discrete Painlevé equations [10,20].…”

Variations of theq-Garnier system

“…There is a convenient method to approach the continuous/discrete Painlevé equations using certain problem of Padé approximation [17,34] (see also [12,13]) or interpolation [6,15,16,18,19,22,36,37]. In [20], this method is applied to the q-Garnier system in the case of section 3.2.1.…”

A Variation of the q-Painlevé System with Affine Weyl Group Symmetry of Type E7(1)

“…In this section we first recall the q-Painlevé type system [28, Section 2.5]. Then we prove that the system is a bi-rational transformation and confirm that the system has the symmetry/surface of type E , and let (f, g) ∈ P 1 ×P 1 be 8 The Padé method has been also applied to the continuous/discrete Painlevé systems in [6,24,25,26,27,30,47,49]. For the case of q-E (1) 7 [24], see Section A.…”

“…, and let (f, g) ∈ P 1 ×P 1 be 8 The Padé method has been also applied to the continuous/discrete Painlevé systems in [6,24,25,26,27,30,47,49]. For the case of q-E (1) 7 [24], see Section A.…”