Meromorphic representations of products of complex arithmetic progressions

Abstract

Given a complex arithmetic sequence, a + nd, where ${a,d \in \mathbb{C}}$, d ≠ 0, and ${n \in \mathbb{Z}^+}$, define ${\Pi^a_d(a+nd):= a(a+d)\cdots (a+nd)}$. At first ${\Pi_d^a}$ is only defined on the terms of the arithmetic sequence. In this article, ${\Pi_d^a}$ is extended to a meromorphic function on ${\mathbb{C} \setminus\{-d,-2d,\dots\}}$ which satisfies the functional equation, ${\Pi_d^a(z+d)=(z+d)\Pi_d^a(z)}$. This extension is represented in three ways: in terms of the classical ${\Pi}$ function; as a limit involving a Pochhammer-type symbol; as an infinite product involving a generalized Euler constant. The infinite product representation leads to a natural Multiplication Formula for the functions ${\Pi_d^a}$, which, in turn, provides an easy way to prove Gauss’s Multiplication Formula for the Γ function.