Quadratic Reduction and Classical Multiple-Index Fibonacci-Lucas Identities

Abstract

We place several classical identities in a common algebraic framework. Let $x$ be an element of a unital algebra over a commutative ring $R$ satisfying $$x^2-tx+d1=0.$$ Then every positive power of $x$ admits the universal reduction $$x^m=P_m(t,d)x-d{P}_{m-1}(t,d)1,$$ where the coefficient polynomials $P_m$ form the generic Lucas sequence and are equivalently Dickson polynomials of the second kind. By the Cayley--Hamilton theorem, this gives the corresponding standard formula for powers of $2\times 2$ matrices, whose scalar coefficients depend only on the trace and determinant. Applying suitable $R$-linear functionals to powers of the Fibonacci matrix yields uniform derivations of multiple-index identities for Fibonacci, Lucas, and generalized Fibonacci sequences. In particular, we recover the expansion for $F_{nm}$ derived by Mc Laughlin, who attributes it to Johnson, and recently reproved by Vorobtsov. The purpose of this note is expository: to make explicit the common mechanism connecting quadratic reduction, Lucas--Dickson polynomials, and matrix methods.

6 pages. v2: Corrected the Chebyshev specialization by including the necessary parity dependence; added the general Dickson-polynomial formulation; clarified the earlier occurrence and derivation of the Fibonacci identity in Mc Laughlin (2004), where it is attributed to Johnson (2003); revised the title and exposition