On $p$-adic incomplete Mellin transforms and $p$-adic incomplete gamma-functions

Let $r$ be a non-zero rational number. In a paper in the Transactions of the AMS in 2023, O'Desky and Richman gave a construction of a $p$-adic incomplete gamma-function $\Gamma_p(\cdot,r)$ for each prime $p$ for which $|r - 1|_p < 1$. Aside from the special case where $r = 1$, only finitely many primes satisfy that condition for a given $r$, so it is desirable to lessen this restriction. In the present paper, we give a construction that works under the much weaker condition that $|r|_p = 1$ using a $p$-adic integral transform we introduced in our paper of 2024 in Acta Arithmetica, which we interpret here as a $p$-adic analogue of an incomplete Mellin transform. For any given $r$, the condition $|r|_p = 1$ holds for all \emph{except} finitely many primes $p$. Our approach emphasizes the parallels between the complex and $p$-adic constructions, explaining how a $p$-adic integration-by-parts formula takes the place of complex integration by parts in the proof of the recurrence relations for the $p$-adic incomplete gamma-functions. We introduce a two-variable $p$-adic transform for the task, extending our earlier $p$-adic integral transform.