### Abstract

We explore a number of functional properties of the q-gamma function and a class of its quotients; including the q-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or asymptotic expansion, together with the fundamental functional equation of the q-gamma function uniquely define those functions. We also study reciprocal "relatives" of the fundamental q-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number π to the q-gamma function. Throughout we highlight the similarities and differences between the cases 0 < q < 1 and q > 1.

Original language | English |
---|---|

Pages (from-to) | 69-110 |

Number of pages | 42 |

Journal | Aequationes Mathematicae |

Volume | 85 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2013 |

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### Keywords

- asymptotic expansion
- complete monotonicity
- multiplication formula
- q-beta function
- q-Gamma functions
- q-reflection formula

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Aequationes Mathematicae*,

*85*(1-2), 69-110. https://doi.org/10.1007/s00010-012-0141-2

**Functional definitions for q-analogues of Eulerian functions and applications.** / ElGuindy, Ahmad; Mansour, Zeinab.

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 85, no. 1-2, pp. 69-110. https://doi.org/10.1007/s00010-012-0141-2

}

TY - JOUR

T1 - Functional definitions for q-analogues of Eulerian functions and applications

AU - ElGuindy, Ahmad

AU - Mansour, Zeinab

PY - 2013

Y1 - 2013

N2 - We explore a number of functional properties of the q-gamma function and a class of its quotients; including the q-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or asymptotic expansion, together with the fundamental functional equation of the q-gamma function uniquely define those functions. We also study reciprocal "relatives" of the fundamental q-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number π to the q-gamma function. Throughout we highlight the similarities and differences between the cases 0 < q < 1 and q > 1.

AB - We explore a number of functional properties of the q-gamma function and a class of its quotients; including the q-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or asymptotic expansion, together with the fundamental functional equation of the q-gamma function uniquely define those functions. We also study reciprocal "relatives" of the fundamental q-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number π to the q-gamma function. Throughout we highlight the similarities and differences between the cases 0 < q < 1 and q > 1.

KW - asymptotic expansion

KW - complete monotonicity

KW - multiplication formula

KW - q-beta function

KW - q-Gamma functions

KW - q-reflection formula

UR - http://www.scopus.com/inward/record.url?scp=84874747236&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874747236&partnerID=8YFLogxK

U2 - 10.1007/s00010-012-0141-2

DO - 10.1007/s00010-012-0141-2

M3 - Article

AN - SCOPUS:84874747236

VL - 85

SP - 69

EP - 110

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 1-2

ER -