[Paper Review] Identities for Fibonacci and Lucas polynomials derived from a book of Gould

Mario Catalani|ArXiv.org|Jul 7, 2004

Advanced Mathematical Theories and Applications2 references4 citations

TL;DR

This paper derives new identities for Fibonacci and Lucas polynomials by leveraging combinatorial identities from Gould's book, particularly those involving binomial coefficients and generating functions. The key contribution is a systematic method to transform these identities into polynomial and numerical results, enabling the derivation of numerous Fibonacci and Lucas number identities through variable substitution.

ABSTRACT

This note is dedicated to Professor Gould. The aim is to show how the identities in his book "Combinatorial Identities" can be used to obtain identities for Fibonacci and Lucas polynomials. In turn these identities allow to derive a wealth of numerical identities for Fibonacci and Lucas numbers.

Motivation & Objective

• To demonstrate how combinatorial identities from Gould's book can be systematically applied to derive new identities for Fibonacci and Lucas polynomials.
• To establish a bridge between generalized polynomial identities and classical numerical sequences like Fibonacci and Lucas numbers.
• To provide a method for generating a wide range of numerical identities by substituting specific values into the derived polynomial identities.
• To explore the structural connections between polynomial recurrence relations and binomial coefficient sums through generating functions and roots of characteristic equations.

Proposed method

• Utilizes known combinatorial identities from Gould's book, particularly those involving binomial coefficients and rational functions, as starting points.
• Applies variable substitutions such as $ z = 4y/x^2 $ and $ z = (eta - eta)/x $ to relate Gould’s identities to polynomial generating functions.
• Employs the Binet formulas for Fibonacci and Lucas polynomials, expressing them in terms of roots $ \alpha $ and $ \beta $ of the characteristic equation $ t^2 - xt - y = 0 $.
• Transforms the right-hand sides of Gould’s identities into expressions involving $ F_n(x,y) $ and $ L_n(x,y) $ using algebraic manipulation and properties of $ \alpha $ and $ \beta $.
• Derives new identities by combining binomial coefficient sums with powers of $ x^2 + 4y $, linking them to polynomial evaluations.
• Uses generating functions $ \frac{t}{1 - xt - yt^2} $ and $ \frac{2 - xt}{1 - xt - yt^2} $ to verify and derive identities systematically.

Experimental results

Research questions

• RQ1 How can combinatorial identities from Gould’s book be adapted to derive identities for bivariate Fibonacci and Lucas polynomials?
• RQ2 What specific substitutions of variables $ x $ and $ y $ yield known numerical identities for Fibonacci and Lucas numbers?
• RQ3 Can identities involving binomial coefficients and rational functions be systematically transformed into polynomial identities using generating functions and roots of characteristic equations?
• RQ4 What structural relationships exist between the polynomial identities and the classical Fibonacci and Lucas number sequences?
• RQ5 How do transformations involving $ \alpha $ and $ \beta $, the roots of the characteristic equation, enable the derivation of new identities?

Key findings

• The identity $ \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k} \frac{n}{n-k} x^{n-2k} y^k = L_n(x,y) $ is derived from Gould’s identity 1.64, linking binomial sums to Lucas polynomials.
• From Gould’s identity 1.38, the identity $ \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} \frac{(x^2 + 4y)^k x^{n-2k}}{2k+1} = \frac{2^n}{n+1} F_{n+1}(x,y) $ is obtained, connecting binomial sums to Fibonacci polynomials.
• The identity $ \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} \left(\frac{x^2 + 4y}{x^2}\right)^k = \frac{2^{n-1}}{x^n} L_n(x,y) $ is derived from identity 1.87, showing a direct link between binomial sums and Lucas polynomials.
• A novel identity $ 2 \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} L_{2k}(x,y) x^{n-2k} = L_n(x,y) + L_n(3x, y - 2x^2) $ is established, revealing a recursive structure involving transformed parameters.
• From identity 1.95, the identity $ \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{2k+1} (x^2 + 4y)^k x^{n-2k-1} = 2^{n-1} F_n(x,y) $ is derived, linking odd binomial sums to Fibonacci polynomials.
• A complex identity involving $ F_{2k}(x,y) $, $ F_{n+1}(x,y) $, and $ F_n(3x, y - 2x^2) $ is derived, showing that $ 2y \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{2k+1} F_{2k}(x,y) x^{n-2k-1} = -[F_{n+1}(x,y) - (y - 2x^2) F_{n-1}(3x, y - 2x^2) - x F_n(3x, y - 2x^2)] $.