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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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