What question did this study set out to answer?

To derive a new class of weighted Fibonacci identities using a matrix-algebraic framework.

July 2, 2026Open Access

Weighted Fibonacci Identities Derived from the Matrix S₄ⁿ (Fₛ, Fₒ+₁)

Key Points

To derive a new class of weighted Fibonacci identities using a matrix-algebraic framework.
Utilized the $4 \times 4$ matrix $S_4^n(F_s, F_{s+1})$ to match closed-form entries with binomial expansions.
Applied basis decomposition and structural convolution kernels to reformulate identities.
Exploited properties such as sparsity and commutativity to develop convolution operators in a persymmetric setting.
Established new combinatorial identities that relate weighted binomial sums to Fibonacci subsequences.
Demonstrated how structural relations in matrices can generate complex identities systematically.

Abstract

This study presents a matrix-algebraic framework for deriving weighted Fibonacci identities from the 4 4 matrix S₄ⁿ (Fₛ, Fₒ+₁). By matching the closed-form entries of this matrix with the corresponding binomial expansions obtained from basis decomposition and structural convolution kernels, a new class of combinatorial identities is derived. These identities relate weighted binomial sums to Fibonacci subsequences of the form F (ₒ+₁) ₍+₊ for k \-1, 0, 1\. The method exploits the sparsity and commutativity of the basis matrices to produce parity-dependent convolution operators in a persymmetric setting. The resulting framework provides a systematic way to generate weighted Fibonacci identities from structured matrix relations.

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