Sums of Squares: Methods for Proving Identity Families

TL;DR

This paper introduces a unified algebraic verification method to prove sum of squares identities for generalized Fibonacci-like sequences, extending known results and suggesting a new trend in identity proofs.

Contribution

It provides a closed-form formula for sums of squares of k-step Fibonacci sequences and introduces a uniform algebraic verification approach for proving such identities.

Findings

Derived a closed formula for sums of squares of generalized Fibonacci sequences.

Presented an algebraic verification method reducing proofs to polynomial equality checks.

Connected the method to existing identities for Fibonacci and Tribonacci numbers.

Abstract

This paper presents both a method and a result. The result presents a closed formula for the sum of the first $m+1,m \ge 0,$ squares of the sequence $F^{(k)}$ where each member is the sum of the previous $k$ members and with initial conditions of $k-1$ zeroes followed by a 1. The generalized result includes the known result of sums of squares of the Fibonacci numbers and a recent result of Schumaker on sums of squares of Tribonacci numbers. To prove the identities uniformly for all $k,$ the Algebraic Verification method is presented which reduces proof of an identity to verification of the equality of finitely many pairs of finite-degree polynomials, possibly in several variables. Several other papers proving families of identities are examined, and it is suggested that the collection of the uniform proof methods used in these papers could produce a new trend in stating and proving identities.