Factorial Transformation for Some Classical Combinatorial Sequences

Abstract

Factorial transformation known from Euler’s time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Padé approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.