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.