Balancing polynomials, Fibonacci numbers and some new series for $\pi$

TL;DR

This paper evaluates infinite series involving balancing and Lucas-balancing polynomials to derive new series for pi that incorporate Fibonacci and Lucas numbers, expanding mathematical understanding of these series.

Contribution

It introduces new closed-form evaluations of series with balancing polynomials, leading to novel pi-related series involving Fibonacci and Lucas numbers.

Findings

Derived new series for pi involving Fibonacci and Lucas numbers

Provided closed-form evaluations of series with balancing polynomials

Extended previous work by Castellanos from 1986 and 1989

Abstract

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $\pi$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.