Expansion Formulas for Multiple Basic Hypergeometric Series Over Root Systems

Abstract: In a series of works, Zhi-Guo Liu extended some of the central summation and transformation formulas of basic hypergeometric series. In particular, Liu extended Rogers' non-terminating very-well-poised $_{6}\phi_{5}$ summation formula, Watson's transformation formula, and gave an alternate approach to the orthogonality of the Askey-Wilson polynomials. These results are helpful in number-theoretic contexts too. All this work relies on three expansion formulas of Liu.

This talk will present several infinite families of extensions of Liu's fundamental formulas to multiple basic hypergeometric series over root systems. We will also discuss results that extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. Subsequently, we will look at an application based on the expansions of infinite products. These extensions have been obtained using the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from $A_n$, $C_n$ and $D_n$ basic hypergeometric series theory. We will observe how this approach brings Liu's expansion formulas within the Bailey transform methodology.

This talk is based on joint work with Gaurav Bhatnagar. (https://arxiv.org/abs/2109.02827)

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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