Abstract
These lecture notes discuss some of the basics of elliptic hypergeometric functions. These are fairly recent generalizations of ordinary hypergeometric functions. In this chapter we first discuss both ordinary hypergeometric functions and elliptic functions, as you need to know both to define elliptic hypergeometric series. We subsequently discuss some of the important properties these series satisfy, in particular we consider the biorthogonal functions found by Spiridonov and Zhedanov, both with respect to discrete and continuous measure. In doing so we naturally encounter the most important evaluation and transformation formulas for elliptic hypergeometric series, and for the associated elliptic beta integral.
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Notes
- 1.
You might ask: What is classical? There are many different definitions based on properties such a classical family should have, which result in different families being called classical or not. As far as I know any definition includes only families from the (q)-Askey scheme, but some are more restrictive. I just use the definition “Everything in the (q-)Askey scheme is classical.”
- 2.
One might observe that the name “contiguous relation” is somewhat of a misnomer. It is not so much that the relation itself is contiguous (which means “next to”) but that it relates functions at parameter values which are contiguous. A better term which is sometimes used, would be “contiguity relation.” However in these notes I prefer to use the terminology of the standard work [1].
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Acknowledgements
I would like to express my gratitude to the organizers of ASIDE 2016 for inviting me to speak there. I would also like to thank an anonymous referee for his many useful remarks.
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van de Bult, F.J. (2017). Elliptic Hypergeometric Functions. In: Levi, D., Rebelo, R., Winternitz, P. (eds) Symmetries and Integrability of Difference Equations. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-56666-5_2
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