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

TL;DR

This paper derives a connection formula for the Jackson integral of type A_n, expressing a bilateral hypergeometric series as a combination of multiple series, and introduces elliptic interpolation functions to explicitly compute connection coefficients.

Contribution

It provides a new explicit connection formula and determinant expression for solutions of q-difference systems related to Jackson integrals of type A_n.

Findings

Derived a connection formula for Jackson integrals of type A_n.

Expressed bilateral hypergeometric series as a linear combination of multiple series.

Established an explicit determinant formula for the fundamental solution matrix.

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.