Borel regularization and Mellin transform of factorial and left factorial series

Abstract

Using Borel summation, we assign finite analytic values to the divergent ordinary generating functions of the factorial sequence n! and the left factorial !n. We consider the Mellin transform of the obtained regularized generating functions and derive a bilateral integral representation for the left factorial. As an application, we prove a closed-form formula for !(−12) – the first explicit analytic expression for the left factorial at a non-integer point.