Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 077, 42 pages      arXiv:1801.07041      https://doi.org/10.3842/SIGMA.2018.077
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

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

Masahiko Ito a and Masatoshi Noumi b
a) Department of Mathematical Sciences, University of the Ryukyus, Okinawa 903-0213, Japan
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received January 23, 2018, in final form July 07, 2018; Published online July 24, 2018

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.

Key words: Jackson integral of type $A_n$; $q$-difference equations; Selberg integral; Slater's transformation formulas; elliptic Lagrange interpolation.

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