Factorial-Type Recurrence Relations and $p$-adic Incomplete Gamma Functions

Abstract

We introduce an automorphism $\mathcal{S}$ of the space $C(\mathbb{Z}_p,\mathbb{C}_p)$ of continuous functions $\mathbb{Z}_p \rightarrow \mathbb{C}_p$ and show that it can be used to give an alternative construction of the $p$-adic incomplete $\Gamma$-functions recently introduced by O'Desky and Richman (arXiv:2012.04615). We then describe various properties of the automorphism $\mathcal{S}$, showing that it is self-adjoint with respect to a certain non-degenerate symmetric bilinear form defined in terms of $p$-adic integration, and showing that its inverse plays a role in a $p$-adic integral-transform space akin to the role of differentiation in the classical space of Laplace-transformed functions. We also derive an integral-transform formula for the $p$-adic incomplete $\Gamma$-functions.