Lucas-Euler relations using balancing and Lucas-balancing polynomials

TL;DR

This paper derives new combinatorial identities involving Euler, Lucas-balancing, Fibonacci, and Euler numbers using elementary techniques and generating functions, expanding the mathematical understanding of these polynomials and their relations.

Contribution

It introduces novel identities connecting Euler and Lucas-balancing polynomials with Fibonacci and Euler numbers, complementing previous Fibonacci-Bernoulli results.

Findings

New identities involving Euler and Lucas-balancing polynomials

Connections established between these polynomials and Fibonacci, Lucas, and Euler numbers

Results derived through elementary methods and functional equations

Abstract

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.