Note on the Degenerate Gamma Function
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Note on the Degenerate Gamma Function T. Kim∗,1 and D. S. Kim∗∗,2 ∗
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea, ∗∗ Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea, E-mail: 1 [email protected], 2 [email protected] Received April 10, 2020; Revised April 15, 2020; Accepted April 17, 2020
Abstract. Recently, the degenerate gamma functions were introduced as a degenerate version of the usual gamma function. In this paper, we investigate several properties of these functions. Namely, we obtain an analytic continuation as a meromorphic function on the whole complex plane, the difference formula, the values at positive integers, some expressions following from the Weierstrass and Euler formulas for the ordinary gamma function, and an integral representation as an integral along a Hankel contour. DOI 10.1134/S1061920820030061
1. INTRODUCTION It is not an exaggeration to say that the gamma function is the most important nonelementary transcendental function. It appears in many areas, such as hypergeometric series, asymptotic series, definite integration, Riemann zeta function, L-functions, and number theory in general. The gamma function was introduced by Euler and subsequently studied by eminent mathematicians like Daniel Bernoulli, Legendre, Gauss, Liouville, Weierstrass, Hermite, as well as many other mathematicians. In [6], the degenerate gamma functions were introduced, as an attempt to find a degenerate version of the ordinary gamma function, and the related degenerate Laplace transform was also studied (see also [7]). In this paper, we would like to derive some basic properties of the degenerate gamma functions including the analytic continuation to a meromorphic function on the whole complex plane, the difference formula, some expressions coming from the Euler and Weierstrass formulas, and an integral representation as an integral along a Hankel contour. We also discuss degenerate beta functions. In the rest of this section, we recall some basic facts on the gamma function, the beta function, the degenerate exponential function, and the degenerate gamma functions. In Sec. 2, we prove the main results of this paper. Finally, we discuss an integral representation of the degenerate gamma functions in Sec. 3 and conclude our results in Sec. 4. Recently, L. M. Upadhyaya published many papers on the degenerate Laplace transforms(see [20]: International Journal of Engineering, Science and Mathematics). However, almost all of his studies are directly derived from the results previously studied by Kim–Kim in [6]. A good example of the application of a gamma function in this paper, is the construction of degenerate gamma random variables used to predict corona viruses (see [11]). It is very difficult to predict the number of people transmitting the virus within a given period of time at a time when the coronavirus is rapidly transmitted in accordance to sudden variables (see [11]). The degenerate gamma function was used to cons
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