[Paper Review] Generalized bivariate Fibonacci polynomials

Mario Catalani|ArXiv.org|Nov 23, 2002

Advanced Mathematical Theories and Applications2 references16 citations

TL;DR

This paper introduces generalized bivariate Fibonacci polynomials as a unifying framework for bivariate Fibonacci and Lucas polynomials, using matrix algebra and Binet-type formulas to derive identities, inequalities, and inversion formulas. The key contribution is a systematic matrix-based derivation of generalized identities, including Simpson's formula and power-sum relations, extending classical results to a broader polynomial family with applications to recurrence relations and eigenvalue analysis.

ABSTRACT

We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in most cases generalize known results.

Motivation & Objective

• To define a generalized bivariate polynomial family that includes Fibonacci and Lucas polynomials as special cases.
• To establish a matrix-based framework for deriving identities and inequalities for these polynomials.
• To generalize classical results such as Binet’s formula, Cassini’s identity, and Simpson’s formula to the bivariate setting.
• To explore inversion formulas and functional transformations, including complex parameter substitutions.
• To analyze eigenvalues and eigenvectors of associated matrices to derive new recurrence and power-sum identities.

Proposed method

• Define the generalized bivariate polynomial $ H_n(x,y) $ via the recurrence $ H_n = xH_{n-1} + yH_{n-2} $ with arbitrary initial conditions $ H_0 = a_0, H_1 = a_1 $.
• Use matrix representation $ extbf{A} = \begin{bmatrix} x & 1 \\ y & 0 \end{bmatrix} $ to express powers of the matrix and derive identities via determinant and trace operations.
• Derive Binet’s formula for $ H_n $ using roots $ \alpha, \beta $ of the characteristic equation $ t^2 - xt - y = 0 $, yielding $ H_n = \frac{(a_1 - \beta a_0)\alpha^n - (a_1 - \alpha a_0)\beta^n}{\alpha - \beta} $.
• Establish the matrix identity $ \textbf{C} \textbf{A}^n = \begin{bmatrix} H_{n+2} & H_{n+1} \\ yH_{n+1} & yH_n \end{bmatrix} $, where $ \textbf{C} = a_0 y \textbf{I} + a_1 \textbf{A} $, to link initial conditions to matrix powers.
• Derive inequalities using quadratic forms and eigenvalues, such as $ \beta^n (z_1^2 + z_2^2) \leq z_1^2 F_{n+1} + z_1 z_2 (1+y) F_n + z_2^2 y F_{n-1} \leq \alpha^n (z_1^2 + z_2^2) $.
• Use Schur’s inequality on matrix elements and eigenvalues to derive bounds, e.g., $ F_{n+1}^2 + (1+y^2)F_n^2 + y^2 F_{n-1}^2 \geq L_{2n} $, with equality when $ y=1 $.

Experimental results

Research questions

• RQ1 How can bivariate Fibonacci and Lucas polynomials be unified under a single generalized polynomial framework with arbitrary initial conditions?
• RQ2 What matrix-based identities can be derived for generalized bivariate polynomials, and how do they extend classical results like Cassini’s or Simpson’s formula?
• RQ3 What functional transformations or inversion formulas exist for these polynomials, particularly under complex parameter substitutions?
• RQ4 How do eigenvalues and eigenvectors of associated matrices relate to the polynomial sequences and their recurrence structures?
• RQ5 What inequalities can be derived from matrix quadratic forms, and how do they generalize known bounds on Fibonacci numbers?

Key findings

• The generalized bivariate polynomial $ H_n(x,y) $ satisfies a Binet-type formula: $ H_n = \frac{(a_1 - \beta a_0)\alpha^n - (a_1 - \alpha a_0)\beta^n}{\alpha - \beta} $, where $ \alpha, \beta $ are roots of $ t^2 - xt - y = 0 $.
• Simpson’s formula is generalized as $ H_n H_{n+2} - H_{n+1}^2 = (-1)^n y^{n-1} (a_0 a_1 x y + a_0^2 y^2 - a_1^2 y) $, reducing to the classical form for Fibonacci and Lucas polynomials.
• The identity $ F_n L_n = F_{2n} $ holds for bivariate Fibonacci and Lucas polynomials, extending the univariate identity.
• An inversion formula is derived: $ F_n(x, -y) = (i\gamma)^{-(n-1)} F_n(i\gamma x, \gamma^2 y) $, with special case $ F_n(x, -y) = (-1)^{n-1} F_n(-x, -y) $ when $ \gamma = i $.
• Power-sum identities are established: $ \sum_{k=0}^n \binom{n}{k} x^k y^{n-k} F_k = F_{2n} $ and $ \sum_{k=0}^n \binom{n}{k} x^k y^{n-k} L_k = L_{2n} $, derived from $ (y + x\alpha)^n + (y + x\beta)^n = \alpha^{2n} + \beta^{2n} $.
• The matrix $ \textbf{BA}^n $ satisfies $ L_n^m = \sum_{k=0}^{\lfloor m/2 \rfloor} \binom{m}{k} (-y)^{kn} L_{n(m-2k)} $ for odd $ m $, and a modified form for even $ m $, generalizing Lucas number power identities.