A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity

Abstract: We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers.

• The paper establishes that if x²−tx+d=0, every power xᵐ reduces to Pₘ(t,d)x−dPₘ₋₁(t,d), where Pₘ is an explicit universal binomial polynomial valid over any commutative ring.
• It connects the reduction formula to Cayley–Hamilton and Chebyshev/Dickson polynomials, showing that powers of any 2×2 matrix depend structurally on its trace and determinant.
• It derives Vorobtsov’s identity Fₙₘ=Fₙ∑ᵢ binom(m−1−i,i)Lₙᵐ⁻¹⁻²ⁱ(−1)ⁱ⁽ⁿ⁺¹⁾, demonstrating that the result follows from a general matrix principle rather than Fibonacci-specific combinatorics.

Overview

This short paper by Marco Mantovanelli establishes a universal reduction formula for powers of elements in quadratic algebras and applies it to derive a binomial expansion for $F_{nm}$, recovering a recent identity of Vorobtsov (Vorobtsov, 12 Mar 2026). The central contribution is conceptual rather than computational: it demonstrates that Fibonacci-specific multiple-index identities are instances of a general algebraic principle governed solely by trace and determinant.

The universal quadratic identity

The main theorem considers an element $x$ of an $R$-algebra over a commutative ring $R$ satisfying the quadratic relation

$x^2 - tx + d = 0.$

Defining polynomials $P_m(t,d)$ via $P_0 = 0$, $P_1 = 1$, and the recurrence $P_{m+1} = tP_m - dP_{m-1}$, the theorem asserts that every power reduces to the span of $\{1, x\}$:

$x$0

with the closed form

$x$1

The proof is elementary: an induction on the coefficient pair $x$2 in the ansatz $x$3, followed by verification that the binomial sum satisfies the same recurrence. A companion proposition upgrades this to a universal statement: since $x$4, the identity holds simultaneously over every commutative ring and every quadratic relation with parameters $x$5. This universality is what licenses the later specialization to integer-valued matrix entries without any case analysis.

The result is essentially a restatement of the Cayley–Hamilton mechanism at the level of an abstract algebra element. Its value lies not in novelty of technique but in packaging: the formula depends only on the two scalar invariants $x$6 and $x$7, which are similarity-invariant.

Matrix formulation and Chebyshev connection

Specializing to $x$8 with $x$9 and $R$0, the Cayley–Hamilton theorem supplies exactly the required quadratic relation, yielding

$R$1

The paper also records the identification of $R$2 with Chebyshev (equivalently Dickson) polynomials: when $R$3 exists in a suitable extension,

$R$4

This recovers the classical Chebyshev representation of second-order recurrences, e.g. $R$5, as a special case. The author notes the square root may not exist in $R$6 itself; the polynomial identity remains valid regardless, which is precisely where the integral binomial form is more robust than the Chebyshev expression.

Application to Fibonacci numbers

Applying the corollary to the Fibonacci matrix $R$7, one uses $R$8 and $R$9. Setting $R$0 so that $R$1, and extracting the $R$2 entry (noting the identity term contributes nothing there), the paper obtains:

$R$3

This is exactly Vorobtsov's identity from (Vorobtsov, 12 Mar 2026). The derivation is immediate—no Fibonacci-specific combinatorial argument is needed—and thereby supports the paper's stated thesis: such identities are consequences of general algebraic principles rather than intrinsic properties of Fibonacci numbers. The same template would apply verbatim to any sequence generated by powers of a $R$4 matrix, including Lucas and generalized Fibonacci sequences, though the paper does not work these cases out explicitly.

Limitations and open questions

The mathematical content is deliberately minimal, and the paper is candid about its scope. The core theorem is a folklore-level consequence of Cayley–Hamilton; the author acknowledges that closely related formulas appear in McLaughlin's study of $R$5 matrix powers (Laughlin, 2018), positioning the contribution as a unifying formulation rather than new machinery. Several points remain open or implicit:

• Generality beyond dimension two: no analogue for elements satisfying relations of degree $R$6 is given, though the recurrence structure suggests one exists.
• Noncommutative coefficients: the framework assumes the quadratic relation has coefficients in the commutative base ring; extensions to skew or parameterized settings are not addressed.
• Combinatorial refinements: the paper derives the identity but does not explore whether the binomial sum admits bijective interpretations, which was part of the motivation in Vorobtsov's original work.
• Scope of applications: only the Fibonacci specialization is computed; parallel identities for Lucas numbers or higher-order recurrences are left unstated.

Conclusion

The paper provides a clean, universal reduction formula for powers in quadratic algebras, expresses it through trace and determinant for $R$7 matrices, and shows that Vorobtsov's binomial identity for $R$8 follows in a few lines as a specialization. Its contribution is a structural clarification: multiple-index Fibonacci identities of this type reflect the Cayley–Hamilton phenomenon rather than arithmetic peculiarities of the Fibonacci sequence.