Topic
q-gamma function
About: q-gamma function is a research topic. Over the lifetime, 80 publications have been published within this topic receiving 2129 citations.
Papers published on a yearly basis
Papers
1 Jan 2005
TL;DR: In this article, q-integral representations of the q-gamma and q-beta functions are studied and a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and Ramanujan's formula for the bilateral hypergeometric series is given.
Abstract: We study q-integral representations of the q-gamma and the q-beta functions This study leads to a very interesting q-constant As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi triple product, including Jacobi's identity, and of Ramanujan's formula for the bilateral hypergeometric series
175 citations
TL;DR: In this article, it was shown that a logarithmically convex function satisfying f(1)=1 and f(x+1)=[(q −1)/(q−1)]f(x) is the q-gamma function.
Abstract: F. H. Jackson defined a generalization of the factorial function by
$$1(1 + q)(1 + q + q^2 ) \cdot \cdot \cdot (1 + q + q^2 + \cdot \cdot \cdot + q^{n - 1} ) = (n!)_q $$
forq>0. He also generalized the gamma function, both for 0 1. Askey then obtained analogues of many of the classical facts about theq-gamma function for 01. It turns out that the log convexity off together with the initial condition and the functional equation no longer forcesf to be theq-gamma function. A stronger condition is needed than the log convexity, and two sufficient conditions are given in this paper. Also we will consider the behavior of theq-gamma function asq-changes forq>1.
52 citations
TL;DR: In this article, sufficient conditions are found for a function involving the gamma function and its reciprocal to be logarithmically completely monotonic, and a decreasing monotonicity of the function is generalized and a known inequality is extended.
Abstract: In this paper, sufficient conditions are found for a function involving the gamma function and its reciprocal to be logarithmically completely monotonic. Consequently, a decreasing monotonicity of the function is generalized and a known inequality is extended.
48 citations
1 Jan 2005
TL;DR: In this paper, three basic properties of logarithmically N-alternating monotonic functions are established and the monotonicity results of some functions involving the gamma and q-gamma functions, which are obtained by W. E. Clark and M. H. Ismail.
Abstract: In the paper, three basic properties of the logarithmically N-alternating
monotonic functions are established and the monotonicity results
of some functions involving the gamma and q-gamma functions, which are obtained
in [W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and
psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129–140.], are generalized
to the logarithmically N-alternating monotonicity.
42 citations
13 Feb 2016
TL;DR: In this paper, the authors study the applicability of the extension of quantum calculus based on two parameters and propose the $(p,q)$-Durrmeyer operators, estimate moments and establish some direct results.
Abstract: In the present paper, we study the applications of the extension of quantum calculus based on two parameters. We define beta function and establish an identity with gamma function, for two parameters $(p,q)$, i e. the post-quantum calculus. We also propose the $(p,q)$-Durrmeyer operators, estimate moments and establish some direct results. Depending on the selection of $p$ and $q,$ the rate of convergence of the our new operators can provide better approximation than those of the Bernstein-Durrmeyer operators and its $q$-analogue. In the end, we provide some graphs using the software Mathematica.
35 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 1 |
| 2019 | 4 |
| 2018 | 6 |
| 2017 | 7 |
| 2016 | 6 |