arXiv 2602.134952026-02-13math.COmath.CAmath.QA

Macdonald超几何级数${}_rΦ_s(x,q,t)$和${}_rΦ_s(x,y,q,t)$通过$q$-差分方程的特征化

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

Rutgers University（罗格斯大学）

Hong Chen

备注：: 24 pages

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AI总结：

本文通过构造$q$-差分算子，特征化了Macdonald超几何级数${}_rΦ_s(x;q,t)$和${}_rΦ_s(x,y;q,t)$。

摘要：

在1980年代广泛流传的两篇手稿中，I.G.Macdonald引入了某些多变量超几何级数${}_pF_q(x;α)$和${}_pF_q(x,y;α)$及其$q$-类比${}_rΦ_s(x;q,t)$和${}_rΦ_s(x,y;q,t)$。这些级数通过Jack和Macdonald多项式给出显式展开，它们推广了来自统计学的一维和二维超几何函数。在最近与Siddhartha Sahi的联合论文中，我们构造了能刻画Jack级数${}_pF_q$的微分算子，从而回答了Macdonald提出的问题。在本文中，我们构造了类似的$q$-差分算子，以刻画Macdonald级数${}_rΦ_s$。更具体地说，我们构造了三个$q$-差分算子$\mathcal A^{(x,y)}$、$\mathcal B^{(x)}$和$C^{(x)}$。方程$\mathcal A^{(x,y)}(f(x,y))=0$刻画${}_rΦ_s(x,y;q,t)$，而方程$\mathcal B^{(x)}(f(x))=0$和$\mathcal C^{(x)}(f(x))=0$各自刻画${}_rΦ_s(x;q,t)$。这些刻画需满足一定的对称性、边界和稳定性条件。在${}_2Φ_1(x;q,t)$的特殊情况下，我们的算子$\mathcal B^{(x)}$于1996年由Kaneko先前构造。

英文摘要：

In two widely circulated manuscripts from the 1980s, I. G. Macdonald introduced certain multivariate hypergeometric series ${}_pF_q(x;α)$ and ${}_pF_q(x,y;α)$ and their $q$-analogs ${}_rΦ_s(x;q,t)$ and ${}_rΦ_s(x,y;q,t)$. 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 ${}_rΦ_s$. 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 ${}_rΦ_s(x,y;q,t)$, while the equations $\mathcal B^{(x)}(f(x))=0$ and $\mathcal C^{(x)}(f(x))=0$ each characterize ${}_rΦ_s(x;q,t)$. These characterizations are subject to certain symmetry, boundary and stability condition. In the special case of ${}_2Φ_1(x;q,t)$, our operator $\mathcal B^{(x)}$ was previously constructed by Kaneko in 1996.

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摘要：

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