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

In this paper, we introduce the (rho, q)-analog of the p-adic factorial function. By utilizing some properties of (rho, q)-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic (rho, 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 (rho, q)-gamma function in general case. Moreover, we consider the p-adic (p, q)-Euler constant derived from the derivation of p-adic (rho, q)-gamma function at x = 1. Furthermore, we provide a limit representation of aforementioned Euler constant based on (rho, q)-numbers. Finally, we consider (rho, q)-extension of the p-adic beta function via the p-adic (rho, q)-gamma function and we then investigate various formulas and identities.