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.

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

Mathematical problems and theory of numerical methods

Varin Victor Petrovich,  ,  orcid.org/0000-0001-7324-1823,  KIAM RAS