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
$$ x^m=P_m(t,d)x-dP_{m-1}(t,d)1, $$
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 $2\times2$ 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.
From: Marco Mantovanelli [view email]
[v1]
Thu, 19 Mar 2026 15:27:51 UTC (4 KB)
[v2]
Tue, 21 Jul 2026 10:14:07 UTC (7 KB)
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