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 $\Gamma $-functions recently introduced by O’Desky and Richman. We then describe various properties of $\mathcal S$, showing in particular that it is self-adjoint with respect to a certain non-degenerate symmetric bilinear form defined in terms of $p$-adic integration, and introducing a $p$-adic integral transform to which $\mathcal S$ is related. We also derive an integral-transform formula for the $p$-adic incomplete $\Gamma $-functions.