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

Let $r$ be a non-zero rational number. In \cite{o’desky-richman:incomplete-gamma}, a construction was given of a $p$-adic incomplete gamma-function $Γ_p(\cdot,r)$ for each prime $p$ for which $|r - 1|_p < 1$. Except in the special case where $r = 1$, only finitely many primes satisfy that condition for a given $r$. 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 \cite{buckingham:factorial}. For any given $r$, this weaker condition holds for all except finitely many primes $p$.

💡 Research Summary

This paper presents a significant generalization of the theory of p-adic incomplete gamma functions. Prior work by O’Desky and Richman constructed a p-adic incomplete gamma function Γ_p(·, r) for a non-zero rational number r, but only for primes p satisfying the restrictive condition |r - 1|_p < 1. For a given r ≠ 1, this condition holds for only finitely many primes. The present work overcomes this limitation by providing a construction that works under the much weaker condition |r|_p = 1. Since, for a fixed r, all but finitely many primes satisfy |r|_p = 1, this new construction defines the p-adic incomplete gamma function for almost all primes, bringing the theory more in line with that of p-adic L-functions.

The core of the interpolation strategy hinges on a global power series with rational coefficients: f_r(t) = -r((1-t)^{1/r} - 1). This series serves as a common seed for both the classical and the p-adic incomplete gamma functions. From f_r(t), one constructs on one hand a C-valued function φ_{f_r}: R≤0 → C, and on the other hand a C_p-valued continuous function φ_{f_r}: Z_p → C_p. The classical incomplete gamma function is obtained from the former via a process akin to an incomplete Mellin transform, while the p-adic version is obtained from the latter via a p-adic integral transform I introduced by the author in prior work.

A key technical innovation is the introduction of a unified 2-variable p-adic transform T_φ(x,y), which generalizes and combines two earlier transforms S and I. This transform is defined as an infinite series involving binomial coefficients and values of φ. Fixing one variable yields families of operators: S_y(φ)(x) = T_φ(x,y) and I_x(φ)(y) = T_φ(x,y). The transform I is identified as a p-adic analogue of the classical incomplete Mellin transform.

The functions derived from f_r(t) via these transforms, both complex and p-adic, satisfy a functional equation originating from the derivative f’_r(t) = (1-t)^{1/r - 1}. This functional equation leads to a recurrence relation on non-negative integers satisfied simultaneously by all these functions. Leveraging this recurrence and the density of the set 1 + (p-1)Z≥0 in Z_p, the paper proves its main interpolation theorem: for integers m in this set, the newly constructed p-adic incomplete gamma function Γ_p(m, r) interpolates the values of the classical incomplete gamma function Γ(m, r) under a suitable embedding τ_p: Q(e) → Q_p.

The paper provides a thorough foundation, reviewing p-adic measures, Mahler series, correspondences between continuous functions and power series (including the Λ isomorphism), and the construction of continuous functions via the p-adic exponential. The result substantially broadens the scope of p-adic incomplete gamma functions and deepens the structural parallels between complex and p-adic analytic number theory.