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On hypergeometric functions and Pochhammer \(k\)-symbol. (English) Zbl 1163.33300
The authors introduce the \(k\)-generalized gamma function \(\Gamma_{k}\) which is one parameter determination of the classical gamma function such that \(\lim_{k\to 1}\Gamma_{k}=\Gamma\). Using the similar technique the \(k\)-generalized beta function \(B_{k}\) and the \(k\)-generalized Pochhammer symbol \((x)_{n,k}\) are introduced. Several identities of these new generalizations are established and integral representations for the \(\Gamma_{k}\) and \(B_{k}\) functions are provided.

MSC:
33B15 Gamma, beta and polygamma functions
33C47 Other special orthogonal polynomials and functions
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