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q-hypergeometric series

In: A Comprehensive Treatment of q-Calculus

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  • Thomas Ernst

    (Uppsala University, Department of Mathematics)

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.

Suggested Citation

  • Thomas Ernst, 2012. "q-hypergeometric series," Springer Books, in: A Comprehensive Treatment of q-Calculus, edition 127, chapter 0, pages 241-277, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-0431-8_7
    DOI: 10.1007/978-3-0348-0431-8_7
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