A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications

TL;DR

This paper introduces a new family of generating functions for reciprocal binomial coefficients, derives integral representations, and applies these functions to evaluate infinite sequences and establish identities involving harmonic and Fibonacci numbers.

Contribution

It presents novel generating functions for reciprocal binomial coefficients, along with integral representations and new identities, expanding tools for analyzing related sequences and numbers.

Findings

Derived integral representations of the generating functions

Established identities connecting reciprocal binomial coefficients with harmonic and Fibonacci numbers

Demonstrated applications in evaluating infinite sequences involving reciprocal binomial coefficients

Abstract

A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained, including identities connecting reciprocal binomial coefficients with harmonic numbers and Fibonacci numbers. The application of the found functions for evaluating infinite numerical sequences involving reciprocal binomial coefficients is demonstrated.