Multivariate analogues of Ruijsenaars' hypergeometric function
Abstract
We construct multivariate analogues of Ruijsenaars' hypergeometric function as nested products of overlap coefficients between eigenfunctions of twisted primitive elements in representations of the modular double of Uq(sl2). The construction is a hyperbolic counterpart of the nested multivariate Askey--Wilson polynomials and functions. Using iterated coproducts, we derive four systems of difference equations for which these multivariate functions are joint eigenfunctions. The four systems correspond to the two sets of variables and the two step sizes iw1 and iw2, which are interchanged by modular duality. We also prove that Ruijsenaars' one-variable transform is unitary in the parameter regime required for the construction, for positive as well as complex conjugate w1,w2. A two-sided normalization of the kernel separates the input and output weights, so that iterating one-variable transforms yields unitary multivariate transforms with explicit inverse and Plancherel formulas.
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BibTeX
@misc{groenevelt2026multivariate,
title = {{Multivariate analogues of Ruijsenaars' hypergeometric function}},
author = {Wolter Groenevelt and Rik Ledoux},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00445},
eprint = {2611.00445},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00445},
}
RIS
TY - UNPB
TI - Multivariate analogues of Ruijsenaars' hypergeometric function
AU - Wolter Groenevelt
AU - Rik Ledoux
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00445
UR - https://arxiv.org/abs/2611.00445
AB - We construct multivariate analogues of Ruijsenaars' hypergeometric function as nested products of overlap coefficients between eigenfunctions of twisted primitive elements in representations of the modular double of U_q(sl_2). The construction is a hyperbolic counterpart of the nested multivariate Askey--Wilson polynomials and functions. Using iterated coproducts, we derive four systems of difference equations for which these multivariate functions are joint eigenfunctions. The four systems correspond to the two sets of variables and the two step sizes iw_1 and iw_2, which are interchanged by modular duality. We also prove that Ruijsenaars' one-variable transform is unitary in the parameter regime required for the construction, for positive as well as complex conjugate w_1,w_2. A two-sided normalization of the kernel separates the input and output weights, so that iterating one-variable transforms yields unitary multivariate transforms with explicit inverse and Plancherel formulas.
ER -