Generalized binomials in fractional calculus

TL;DR

This paper introduces a class of generalized binomials in fractional calculus, explores their properties, identities, generating functions, and asymptotic behaviors, expanding the mathematical tools available for fractional calculus applications.

Contribution

It provides new combinatorial identities, generating function formulations, and an asymptotic Binomial Theorem for generalized binomials in fractional calculus.

Findings

Derived combinatorial identities including Pascal's rule adaptation

Established recursive, combinatorial, and integral generating functions

Proved an asymptotic version of the Binomial Theorem

Abstract

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities, including an adapted version of the Pascal's rule. We then investigate the associated generating functions, for which we establish a recursive, combinatorial and integral formulation. From this, we derive an asymptotic version of the Binomial Theorem. A combinatorial and asymptotic analysis of some finite sums completes the paper.