A Characterization of the Macdonald Hypergeometric Series ${}_rΦ_s(x;q,t)$ and ${}_rΦ_s(x,y;q,t)$ via $q$-Difference Equations
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866912904303869952 |
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| author | Chen, Hong |
| author_facet | Chen, Hong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13495 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Characterization of the Macdonald Hypergeometric Series ${}_rΦ_s(x;q,t)$ and ${}_rΦ_s(x,y;q,t)$ via $q$-Difference Equations Chen, Hong Combinatorics Classical Analysis and ODEs Quantum Algebra 05E05, 33C67, 39A13 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. |
| title | A Characterization of the Macdonald Hypergeometric Series ${}_rΦ_s(x;q,t)$ and ${}_rΦ_s(x,y;q,t)$ via $q$-Difference Equations |
| topic | Combinatorics Classical Analysis and ODEs Quantum Algebra 05E05, 33C67, 39A13 |
| url | https://arxiv.org/abs/2602.13495 |