A Characterization of the Macdonald Hypergeometric Series ${}_r\Phi_s(x;q,t)$ and ${}_r\Phi_s(x,y;q,t)$ via $q$-Difference Equations

Abstract

In two widely circulated manuscripts from the 1980s, I. G. Macdonald introduced certain multivariate hypergeometric series ${}_pF_q(x;\alpha)$ and ${}_pF_q(x,y;\alpha)$ and their $q$-analogs ${}_r\Phi_s(x;q,t)$ and ${}_r\Phi_s(x,y;q,t)$. These series are given by explicit expansions in Jack and Macdonald polynomials, and they generalize the hypergeometric functions of one and two matrix arguments from statistics. In a recent joint paper with Siddhartha Sahi, we constructed differential operators that characterize the Jack series ${}_pF_q$ thereby answering a question of Macdonald. In this paper we construct analogous $q$-difference operators that characterize the Macdonald series ${}_r\Phi_s$. More precisely, we construct three $q$-difference operators $\mathcal A^{(x,y)}$, $\mathcal B^{(x)}$, $C^{(x)}$. The equation $\mathcal A^{(x,y)}(f(x,y))=0$ characterizes ${}_r\Phi_s(x,y;q,t)$, while the equations $\mathcal B^{(x)}(f(x))=0$ and $\mathcal C^{(x)}(f(x))=0$ each characterize ${}_r\Phi_s(x;q,t)$. These characterizations are subject to certain symmetry, boundary and stability condition. In the special case of ${}_2\Phi_1(x;q,t)$, our operator $\mathcal B^{(x)}$ was previously constructed by Kaneko in 1996.