More Fibonacci-Bernoulli relations with and without balancing polynomials

TL;DR

This paper explores new relationships between Bernoulli polynomials, Fibonacci numbers, and balancing polynomials, deriving combinatorial identities and special cases that extend previous work in the field.

Contribution

It introduces novel identities linking Bernoulli polynomials with Fibonacci and Lucas numbers, generalizing earlier results and providing new insights into their interrelations.

Findings

Derived new combinatorial identities involving Bernoulli and Fibonacci numbers

Proved a special identity connecting Bernoulli polynomials with Fibonacci numbers in arithmetic progression

Extended and generalized previous results by Frontczak (2019)

Abstract

We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers. Moreover, we prove a special identity involving Bernoulli polynomials and Fibonacci numbers in arithmetic progression. Special cases and some corollaries will highlight interesting aspects of our findings. Our results complement and generalize these of Frontczak (2019).