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

Ugur Duran,Mehmet Acikgoz

doi:10.3390/mca24020053

Abstract

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.

Highlights

The p-adic numbers are a counterintuitive arithmetic system, which were firstly introduced by Kummer in 1850
We provide a limit representation of aforementioned Euler constant based on (ρ, q)-numbers
The p-adic numbers are less well known than the others; these numbers play a main role in number theory and the related topics in mathematics

Summary

Introduction

The p-adic numbers are a counterintuitive arithmetic system, which were firstly introduced by Kummer in 1850. We consider (ρ, q)-extension of the p-adic beta function via the p-adic (ρ, q)-gamma function and we investigate various formulas and identities. The q-extension of the p-adic gamma function is defined as follows (see [12]) We provide a new definition of p-adic (ρ, q)-gamma function and gives some properties, identities and relations for the mentioned gamma function.

Results

Conclusion