A System of Four simultaneous Recursions: Generalization of the Ledin-Shannon-Ollerton Identity

TL;DR

This paper generalizes a mathematical identity related to metallic sequences, providing closed-form formulas for sums involving these sequences and integers, through solving a system of four recursions and introducing new OEIS sequences.

Contribution

It extends the Ledin-Shannon-Ollerton identity to all metallic sequences using a novel approach involving four simultaneous recursions and polynomial analysis.

Findings

Derived closed-form formulas for sums involving metallic sequences.

Reduced proof complexity by solving a system of four recursions.

Identified new OEIS sequences from polynomial coefficients.

Abstract

This paper further generalizes a recent result of Shannon and Ollerton who resurrected an old identity due to Ledin. This paper generalizes the Ledin-Shannon-Ollerton result to all the metallic sequences. The results give closed formulas for the sum of products of powers of the first $n$ integers with the first $n$ members of the metallic sequence. Three key innovations of this paper are i) reducing the proof of the generalization to the solution of a system of 4 simultaneous recursions; ii) use of the shift operation to prove equality of polynomials; and iii) new OEIS sequences arising from the coefficients of the four polynomial families satisfying the 4 simultaneous recursions.