Some Properties of the Generalization of the p-Adic Factorial and the p-Adic Gamma Function

In this paper, we introduce the concept of q-adic factorial, establishing several significant congruences and inequalities associated with this new function. For a good illustration, we provide some numerical examples. Furthermore, we prove several properties of the generalized p-adic gamma function Γ_q, along with identities and special values related to Γ_q. We also derive a representation of Γ_q via its Mahler expansion and establish relationships among the coefficients within this expansion.