A Characterization of Macdonald's Jack Hypergeometric Series ${}_pF_q(x;α)$ and ${}_pF_q(x,y;α)$ via Differential Equations

TL;DR

This work develops a comprehensive differential-equation framework for characterizing Macdonald's Jack hypergeometric series ${}_pF_q(x;\alpha)$ and ${}_pF_q(x,y;\alpha)$ in multiple alphabets. By constructing four differential operators ${}\mathcal{A}^{(x,y)}$, ${}\mathcal{B}^{(x)}$, ${}\mathcal{C}^{(x)}$, together with lowering/raising mechanisms ${}\mathcal{L}$ and ${}\mathcal{R}$ built from iterated commutators with the Laplace–Beltrami operator, the authors prove that the two-alphabet series is the unique solution to ${}\mathcal{A}^{(x,y)}F=0$ and the one-alphabet series to ${}\mathcal{C}^{(x)}F=0$ (with symmetry and initial conditions). A stability condition is shown necessary for uniqueness of the one-alphabet solution when using the lowering operator; the raising-operator approach provides an alternative route to a clean uniqueness result (Theorem C). These results extend previous cases known for limited $(p,q)$ by Macdonald, Yan, Kaneko, Baker–Forrester, Muirhead, and Fujikoshi, and unify them under a generating-function viewpoint for eigenvalues. The framework paves the way for q-analogs, Macdonald-polynomial variants, and non-symmetric extensions, with potential applications in multivariate statistics and representation theory.

Abstract

In a widely circulated manuscript from the 1980s, now available on the arXiv, I.~G.~Macdonald introduced certain multivariable hypergeometric series ${}_pF_q(x)= {}_pF_q(x;α)$ and ${}_pF_q(x,y)= {}_pF_q(x,y;α)$ in one and two sets of variables $x=(x_1,\dots x_n)$ and $y=(y_1,\dots y_n)$. These two series are defined by explicit expansions in terms of Jack polynomials $J^{(α)}_λ$, and for $α=2$ they specialize to the hypergeometric series of matrix arguments studied by Herz (1955) and Constantine (1963) that admit analogous expansions in terms of zonal polynomials. In this paper we determine explicit partial differential equations that characterize ${}_pF_q$, thereby answering a question posed by Macdonald. More precisely, for each $n,p,q$ we construct three differential operators $\mathcal A=\mathcal A^{(x,y)}$, $\mathcal B=\mathcal B^{(x)}$, $\mathcal C=C^{(x)}$, and we show that ${}_pF_q(x,y)$ and ${}_pF_q(x)$ are the unique series solutions of the equations $\mathcal A(f)=0$ and $\mathcal C(f)=0$, respectively, subject to certain symmetry and boundary conditions. We also prove that the equation $\mathcal B(f)=0$ characterizes ${}_pF_q(x)$, but only after one restricts the domain of $\mathcal B$ to the set of series satisfying an additional stability condition with respect to $n$. Special cases of the operators $\mathcal A$ and $\mathcal B$ have been constructed previously in the literature, but only for a small number of pairs $(p,q)$, namely for $p \leq 3$ and $q \leq 2$ in the zonal case by Muirhead (1970), Constantine--Muirhead (1972), and Fujikoshi (1975); and for $p \leq 2$ and $q \leq 1$ in the general Jack case by Macdonald (1980s), Yan (1992), Kaneko (1993), and Baker--Forrester (1997). However the operator $\mathcal C$ seems to be new even for these special cases.

Figures (1)

• Figure 1: Previously known differential equations for ${}_{p}F_{q}$

Theorems & Definitions (31)

• Lemma 1
• proof
• Example 1
• Example 2
• Proposition 1
• proof
• Theorem A
• proof
• Lemma 2
• proof