Difference equations and Omega functions

TL;DR

This paper introduces Omega functions, generalizations of Euler Gamma functions, and explores their properties, solutions to their functional equations, and their role as exponential periods with meromorphic extensions.

Contribution

It defines Omega functions as solutions to a specific difference equation, characterizes their solution space, and introduces Incomplete Omega functions for proofs.

Findings

Omega functions satisfy a specific functional difference equation.

The solution space of the equation is finite dimensional and spanned by Omega functions.

Omega functions have a meromorphic extension with simple poles at negative integers.

Abstract

We introduce Omega functions that generalize Euler Gamma functions and study the functional difference equation they satisfy. Under a natural exponential growth condition, the vector space of meromorphic solutions of the functional equation is finite dimensional. We construct a basis of the space of solutions composed by Omega functions. Omega functions are defined as exponential periods. They have a meromorphic extension to the complex plane of order 1 with simple poles at negative integers. The vector space they span is characterized by their functional equation and their growth property on vertical strips. This generalizes Wielandt's characterization of Euler Gamma function. We also introduce Incomplete Omega functions that play an important role in the proofs.