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 Pade approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express
integral representations of them in a closed form.

Keywords:

factorial transformation; factorial series; continued fractions; Stirling, Bernoulli, Bell, Euler and Tangent numbers; divergent power series; generating functions

Publication language:english,
pages:31

Research direction:

Mathematical problems and theory of numerical methods