Some Classes of series involving the Riemann zeta function, Fibonacci numbers and the Lucas numbers

TL;DR

This paper derives explicit formulas for various infinite series involving Fibonacci and Lucas numbers combined with the Riemann zeta function, using classical methods and sequence generating functions.

Contribution

It provides new explicit expressions for series involving Fibonacci, Lucas, and the Riemann zeta function, enhancing understanding of their interrelations.

Findings

Explicit formulas for series involving Fibonacci and Lucas numbers with the Riemann zeta function

Use of Binet formulas and generating functions to derive series expressions

Connections between special sequences and the Riemann zeta function elucidated

Abstract

The objective of this manuscript is to offer explicit expressions for diverse categories of infinite series incorporating the Fibonacci (Lucas) sequence and the Riemann zeta function. In demonstrating our findings, we will utilize conventional methodologies and integrate the Binet formulas pertinent to these sequences with generating functions that encompass the Riemann zeta function alongside established evaluations of certain series.