# A Comprehensive Treatment of q-Calculusq-hypergeometric series

A Comprehensive Treatment of q-Calculus: q-hypergeometric series [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 Γq 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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Comprehensive Treatment of q-Calculusq-hypergeometric series

37 pages      /lp/springer-journals/a-comprehensive-treatment-of-q-calculus-q-hypergeometric-series-s2XIo6HKNZ
Publisher
Springer Basel
ISBN
978-3-0348-0430-1
Pages
241 –277
DOI
10.1007/978-3-0348-0431-8_7
Publisher site
See Chapter on Publisher Site

### 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 Γq 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.]

Published: Jun 18, 2012

Keywords: Tilde Operator; Hypergeometric Series; Summation Formula; Umbral Method; Balanced Quotient