Sums of Squares: Methods for Proving Identity Families

2021/03/31 by Russell Jay Hendel, Hendel, Russell Jay
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT) #Primary: 11B39 Secondary: 11B37 11C99 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2103.16756

openalex publication_date 2021/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≥ 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.

Related

• Construction of Identities for Sums of Squares
• Finite sums associated with some polynomial identities
• On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials
• Squares of Tribonacci numbers
• Sums of integers and sums of their squares