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Open AccessArticle

A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

by Ugur Duran 1,* and Mehmet Acikgoz 2
1
Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, İskenderun Technical University, Hatay TR-31200, Turkey
2
Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep TR-27310, Turkey
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(2), 53; https://doi.org/10.3390/mca24020053
Received: 23 April 2019 / Revised: 19 May 2019 / Accepted: 20 May 2019 / Published: 21 May 2019
(This article belongs to the Section Natural Sciences)
In this paper, we introduce the ρ , q -analog of the p-adic factorial function. By utilizing some properties of ρ , q -numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic ρ , q -gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2 . We also derive more representations of the p-adic ρ , q -gamma function in general case. Moreover, we consider the p-adic ρ , q -Euler constant derived from the derivation of p-adic ρ , q -gamma function at x = 1 . Furthermore, we provide a limit representation of aforementioned Euler constant based on ρ , q -numbers. Finally, we consider ρ , q -extension of the p-adic beta function via the p-adic ρ , q -gamma function and we then investigate various formulas and identities. View Full-Text
Keywords: p-adic numbers; p-adic factorial function; p-adic gamma function; p-adic beta function; p-adic Euler constant; (ρ, q)-numbers p-adic numbers; p-adic factorial function; p-adic gamma function; p-adic beta function; p-adic Euler constant; (ρ, q)-numbers
MDPI and ACS Style

Duran, U.; Acikgoz, M. A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions. Math. Comput. Appl. 2019, 24, 53.

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