Abstract
In this paper, some new integrals involving k gamma function and k digamma function have been established. An integral is established involving k gamma function, and its special values are discussed. Similarly, some new integrals have been established for k digamma function, and different elementary function is associated with it for different values of k. A nice representation of the Euler-Mascheroni constant and π in the form of k digamma function for different values of k is also obtained.
Highlights
In this paper, some new integrals involving k gamma function and k digamma function have been established
Mathematics subject classification: 33B15, 41A58, 33C20 k gamma function The k gamma function is a generalization of the classical gamma function introduced by Diaz and Pariguan [1], denoted and defined as n!k n ðnk
A simple change of variable tk = ky reveals the relationship between k gamma function and classical gamma function
Summary
The symbol for (z)n, k is called Pochhammer’s k symbol [2] and is defined as ðzÞn;k 1⁄4 zðz þ kÞðz þ 2kÞ⋯ðz þ ðn − 1ÞkÞ: ð1:2Þ. A simple change of variable tk = ky reveals the relationship between k gamma function and classical gamma function. In the “An integral involving the k gamma function” section, we will establish an integral involving k gamma function and its special cases will be discussed. In the “Stirling formula for the k gamma function” section, the Stirling formula will be derived for the k gamma function. In the “Some integrals representing digamma function” section, we will provide few integrals involving the k digamma function. In the “Euler-Mascheroni constant and k digamma function” section, we will find the relationship between the EulerMascheroni constant in the form k digamma function for different values of k. An integral involving the k gamma function we will derive an interesting integral involving k gamma function. Njpjkz kz k zθ cos k e − aun ð cosbunÞunkz − 1du; ð2:1aÞ sinzkθ 1⁄4 Z∞
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