The holonomic triangle: from a symmetry between $e$ and $\pi$ to additive Gamma functions

TL;DR

This paper introduces additive Gamma functions (AGFs), a new class of functions satisfying additive functional equations, revealing a symmetry between constants e and pi, and unifying discrete and continuous mathematical structures.

Contribution

It defines AGFs, explores their structural dichotomy, and establishes a holonomic triangle linking P-recursive sequences, additive functional equations, and differential equations.

Findings

AGFs include Euler's Gamma as the order-1 case

Two types of AGFs: regular (Gamma ratios) and irregular (involving incomplete Gamma)

AGFs unify discrete and continuous mathematical frameworks

Abstract

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational coefficients. We call such functions additive Gamma functions (AGFs), recognizing Euler's $\Gamma(z)$ as the order-1 prototype. Our theory reveals a structural dichotomy: one AGF is expressible as Gamma ratios (regular case), another involves incomplete Gamma (irregular case). AGFs complete a holonomic triangle between P-recursive sequences, additive functional equations, and differential equations, unifying discrete and continuous perspectives under the condition that Gamma factors in asymptotics have integer slopes.