Integration Formulas Involving Fibonacci and Lucas Numbers

TL;DR

This paper derives complex integration formulas involving Fibonacci and Lucas numbers, connecting them with special functions and extending previous integral identities.

Contribution

It introduces new integration formulas involving Fibonacci and Lucas numbers, utilizing a fundamental lemma on differentiation of complex-valued Fibonacci and Lucas functions.

Findings

Derived new integration formulas involving Fibonacci and Lucas numbers

Connected Fibonacci and Lucas integrals with special functions like dilogarithm and Clausen's function

Extended previous integral identities by Dilcher and Stewart

Abstract

We present a range of difficult integration formulas involving Fibonacci and Lucas numbers and trigonometric functions. These formulas are often expressed in terms of special functions like the dilogarithm and Clausen's function. We also prove complements of integral identities of Dilcher (2000) and Stewart (2022). Many of our results are based on a fundamental lemma dealing with differentiation of complex-valued Fibonacci (Lucas) functions.