Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# 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.

Journal reference: Mathematical and Cmputational Applications 2019, 24, 53
DOI: 10.3390/mca24020053

## 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.

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