Relating certain weighted Fibonacci series to Bernoulli polynomials via the polylogarithm function

TL;DR

This paper establishes a connection between weighted Fibonacci and Lucas series and polylogarithms, providing explicit expressions involving Bernoulli polynomials in special cases, thereby linking these mathematical concepts.

Contribution

It introduces a novel method to express weighted Fibonacci and Lucas series as linear combinations of polylogarithms, and evaluates some cases using Bernoulli polynomials.

Findings

Weighted Fibonacci and Lucas series can be expressed as polylogarithm combinations.

Special cases allow explicit evaluation using Bernoulli polynomials.

The work links Fibonacci series, polylogarithms, and Bernoulli polynomials.

Abstract

We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between these polynomials and the polylogarithm.