Connection Formula for the Jackson Integral of Type A_n and Elliptic Lagrange Interpolation

Abstract

We investigate the connection problem for the Jackson integral of type A n . Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coefficients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of q-difference equations.

q-Difference Systems for the Jackson Integral of Symmetric Selberg Type

“…Section 5 is devoted to the proof of Lemma 4.1. In Section 6 we explain the Gauss decomposition of the transition matrix R. For this purpose, we introduce another set of symmetric polynomials called the Lagrange interpolation polynomials of type A in [19], which are different from Matsuo's polynomials. Both upper and lower triangular matrices in the decomposition can be understood as a transition matrix between Matsuo's polynomials and the other polynomials.…”

“…We remark that the polynomials f r (a 1 , a 2 ; t; z) are called the Lagrange interpolation polynomials of type A and their properties are discussed in [19,Appendix B]. By definition the polynomial f i (a 1 , a 2 ; t; z) satisfies f i (a 1 , a 2 ; t; z) = f n−i (a 2 , a 1 ; t; z).…”

“…for convergence of the Jackson integrals (1.11). (See [19,Lemma 3.1] for details of convergence.) For the polynomials (1.10) let R be the (n + 1) × (n + 1) matrix defined by e n (a 2 , b 1 ; z), e n−1 (a 2 , b 1 ; z), .…”

$q$-Difference Systems for the Jackson Integral of Symmetric Selberg Type

“…for convergence of the Jackson integrals (1.12). (See Lemma 3.1 in [9] for details of convergence. )…”

“…Section 5 is devoted to the proof of Lemma 4.1. In Section 6 we explain the Gauss decomposition of the transition matrix R. For this purpose, we introduce another set of symmetric polynomials called the Lagrange interpolation polynomials of type A in the context [9], which are different from Matsuo's polynomials. Both upper and lower triangular matrices in the decomposition can be understood as a transition matrix between Matsuo's polynomials and the other polynomials.…”