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

This study presents a matrix-algebraic framework for deriving weighted Fibonacci identities from the 4 × 4 matrix S₄ⁿ(Fₛ, Fₛ₊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₍ₛ₊₁₎ₙ₊ₖ 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.