Elliptic beta integrals and special functions of hypergeometric type (Q2758429)

This paper provides a short review of some recent results by the author on elliptic or modular hypergeometric series [\textit{V. P. Spiridonov}, Russ. Math. Surv. 56, 185-186 (2001; Zbl 0997.33009); An elliptic beta integral, Proc. Fifth Int. Conf. on Diff. Eq. and Appl., to appear]. After a general introduction the first theorem stated is an elliptic analogue of the important Nassrallah-Rahman integral [\textit{B. Nassrallah} and \textit{M. Rahman}, SIAM J. Math. Anal. 16, 186-197 (1985; Zbl 0564.33009); \textit{M. Rahman}, Can. J. Math. 38, 605-618 (1986; Zbl 0599.33015)] which itself is a generalization of the well-known Askey-Wilson integral [\textit{R. Askey} and \textit{J. Wilson}, Mem. Am. Math. Soc. 319, 1-55 (1985; Zbl 0572.33012)] NEWLINENEWLINENEWLINEThe next Theorem of the paper provides an elliptic version of Rahman's continuous biorthogonal rational functions. NEWLINENEWLINENEWLINEFinally, a most intriguing theorem is presented, that appears to be given here for the first time. It is shown that instead of taking the in a sense straightforward elliptic analogue of Rahman's result, one can actually achieve more at the elliptic level. That is, Rahman's \(_{10}\phi_9\) basic hypergeometric series may, only at the elliptic level, be replaced by the product of two such series, one with base \(q\) and one with base \(p\), reflecting the symmetry in these two modular parameters. Two appropriately chosen such products of elliptic \(_{10}\phi_9\)'s are then shown to again form a biorthogonal system. It is also noted that this same product of elliptic \(_{10}\phi_9\)'s admits an integral representation, which again does not exist at the trigonometric level. NEWLINENEWLINENEWLINEReviewer's remark: Many of the more elementary results for basic hypergeometric series such as, for example, the \(q\)-binomial theorem and the Jacobi triple product identity, do not carry over the elliptic case. It is thus most interesting to see there is another side of the story: there exist results that hold at the level of elliptic hypergeometric series only.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00056].