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A Comprehensive Treatment of q-Calculus pp 241–277Cite as

Birkhäuser

q-hypergeometric series

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Abstract

This chapter starts with the general definition of q-hypergeometric series. This definition contains the tilde operator and the symbol ∞, dating back to the year 2000. The notation △(q;l;λ), a q-analogue of the Srivastava notation for a multiple index, plays a special role. We distinguish different kind of parameters (exponents etc.) by the | sign. A new phenomenon is that we allow q-shifted factorials that depend on the summation index. We follow exactly the structure of the definitions in Section 3.7. We quote a theorem of Pringsheim about the slightly extended convergence region compared to the hypergeometric series. Many well-known formulas with proofs are given in the new notation, i.e. the Bayley-Daum summation formula is given with a right-hand side which only contains Γ q functions. This has the advantage that we immediately can compute the limit q→1. Finally, we present three q-analogues of Euler’s integral formula for the Γ function.

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  1. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Thomas Ernst

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Ernst, T. (2012). q-hypergeometric series. In: A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0431-8_7

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