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Multivariable Askey–Wilson function and bispectrality

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Abstract

For every positive integer d, we define a meromorphic function F d (n;z), where n,z∈ℂd, which is a natural extension of the multivariable Askey–Wilson polynomials of Gasper and Rahman (Theory and Applications of Special Functions, Dev. Math., vol. 13, pp. 209–219, Springer, New York, 2005). It is defined as a product of very-well-poised 8 φ 7 series and we show that it is a common eigenfunction of two commutative algebras \({\mathcal{A}}_{z}\) and \({\mathcal{A}}_{n}\) of difference operators acting on z and n, with eigenvalues depending on n and z, respectively. In particular, this leads to certain identities connecting products of very-well-poised 8 φ 7 series.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332–0160, USA

    Jeffrey S. Geronimo & Plamen Iliev

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  1. Jeffrey S. Geronimo
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  2. Plamen Iliev
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Correspondence to Plamen Iliev.

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The work of the second author is partially supported by NSF grant DMS-0901092.

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Geronimo, J.S., Iliev, P. Multivariable Askey–Wilson function and bispectrality. Ramanujan J 24, 273–287 (2011). https://doi.org/10.1007/s11139-010-9244-3

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  • DOI: https://doi.org/10.1007/s11139-010-9244-3

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