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

We introduce an automorphism $\mathcal{S}$ of the space $C({\mathbb{Z}}_p,{\mathbb{C}}_p)$ of continuous functions ${\mathbb{Z}}_p\to {\mathbb{C}}_p$ and show that it can be used to give an alternative construction of the $p$-adic incomplete $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-functions.