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 ${}_p{F}_q(x;\alpha )$ and ${}_p{F}_q(x,y;\alpha )$ 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 ${}_p{F}_q$ 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.

24 pages