q-hypergeometric series

Thomas Ernst
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Thomas Ernst: Uppsala University, Department of Mathematics

Chapter Chapter 7 in A Comprehensive Treatment of q-Calculus, 2012, pp 241-277 from Springer

Abstract: 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.

Keywords: Tilde Operator; Hypergeometric Series; Summation Formula; Umbral Method; Balanced Quotient (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0431-8_7

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DOI: 10.1007/978-3-0348-0431-8_7

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