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A Study on Novel Extensions for the $p$-adic Gamma and $p$-adic Beta Functions
Version 1
: Received: 20 May 2019 / Approved: 21 May 2019 / Online: 21 May 2019 (11:37:44 CEST)
A peer-reviewed article of this Preprint also exists.
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. 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.
Abstract
In this paper, we introduce the (ρ,q)-analogue 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.
Keywords
p-adic numbers; p-adic factorial function; p-adic gamma function; p-adic beta function; p-adic Euler constant; (ρ,q)-numbers
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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