[Paper Review] Additional close links between balancing and Lucas-balancing polynomials

Robert Frontczak, Taras Goy|arXiv (Cornell University)|Jul 28, 2020

Advanced Combinatorial Mathematics12 references4 citations

TL;DR

This paper derives new identities for balancing and Lucas-balancing polynomials using ordinary generating functions, revealing deep connections to Chebyshev polynomials, Fibonacci, and Lucas numbers. Key results include novel combinatorial identities involving these sequences, especially when evaluated at specific points, and closed-form relations linking the polynomials through generating function manipulation and Binet-type formulas.

ABSTRACT

Using generating functions, we derive many identities involving balancing and Lucas-balancing polynomials. By relating these polynomials to Chebyshev polynomials of the first and second kind, and Fibonacci and Lucas numbers, we offer some presumably new combinatorial identities involving these famous sequences.

Motivation & Objective

• To establish new algebraic and combinatorial identities involving balancing and Lucas-balancing polynomials using generating functions.
• To explore connections between these polynomials and classical orthogonal polynomials such as Chebyshev polynomials of the first and second kind.
• To derive identities that specialize to known results involving Fibonacci and Lucas numbers when evaluated at specific values.
• To unify and extend previous results on finite product sums and derivative relations of these polynomials.
• To present new combinatorial identities through evaluation of polynomial identities at critical points involving roots of unity and Lucas numbers.

Proposed method

• Derivation of ordinary generating functions for $ B_n(x) $, $ C_n(x) $, and their even- and odd-indexed subsequences using a known lemma on linear recurrence generating functions.
• Use of Binet’s formulas for $ B_n(x) $ and $ C_n(x) $, involving $ \lambda(x) = 3x + \sqrt{9x^2 - 1} $, to verify identities algebraically.
• Application of generating function manipulation to derive recurrence-like identities between $ B_n(x) $, $ C_n(x) $, and their even/odd indexed components.
• Evaluation of identities at $ x = \frac{\varepsilon_n}{6}L_s $, where $ \varepsilon_n $ is $ 1 $ or $ i $, to generate identities involving Fibonacci and Lucas numbers.
• Use of the golden ratio $ \alpha = \frac{1+\sqrt{5}}{2} $, $ \beta = \frac{1-\sqrt{5}}{2} $, to express identities in terms of Fibonacci and Lucas number sequences.
• Derivation of symmetric identities involving binomial coefficients, powers of $ \frac{\sqrt{5}F_s}{2} $, and alternating signs to relate sums of Fibonacci and Lucas numbers.

Experimental results

Research questions

• RQ1 How can generating functions be used to derive new identities between balancing and Lucas-balancing polynomials?
• RQ2 What connections exist between balancing polynomials and Chebyshev polynomials of the first and second kind?
• RQ3 How do evaluations of polynomial identities at $ x = \frac{\varepsilon_n}{6}L_s $ yield new combinatorial identities for Fibonacci and Lucas numbers?
• RQ4 What symmetric identities emerge from binomial-weighted sums of Fibonacci and Lucas numbers with alternating signs and powers of $ \frac{\sqrt{5}F_s}{2} $?
• RQ5 Can the structure of $ B_n(x) $ and $ C_n(x) $ be used to derive identities involving $ F_{sn} $, $ L_{sn} $, and their products?

Key findings

• The identity $ B_n(x) - 3x B_{n-1}(x) = C_{n-1}(x) $ holds for all $ n \geq 1 $, establishing a direct algebraic link between the two polynomial families.
• The relation $ 3x(B_{2n+1}(x) - B_{2n-1}(x)) = C_{2n+1}(x) + C_{2n-1}(x) $ is derived via generating function analysis and verified using Binet’s formula.
• Evaluating at $ x = \frac{\varepsilon_n}{6}L_s $, the identity $ C_n\left(\frac{\varepsilon_n}{6}L_s\right) = \varepsilon_n^n \frac{L_{sn}}{2} $ is established, linking Lucas-balancing polynomials to Lucas numbers.
• The identity $ 2F_{sn} = F_s L_{s(n-1)} + L_s F_{s(n-1)} $ is derived as a special case of the general polynomial identities.
• A symmetric identity involving binomial coefficients and powers of $ \frac{\sqrt{5}F_s}{2} $ is proven: $ \sqrt{5}\sum_{k=1}^n \binom{n}{k} \left(\frac{\sqrt{5}F_s}{2}\right)^{n-k}(1+(-1)^{n-k})F_{ks} = \sum_{k=0}^n \binom{n}{k} \left(\frac{\sqrt{5}F_s}{2}\right)^{n-k}(1-(-1)^{n-k})L_{ks} $.
• The paper establishes that $ F_{2sn} - (-1)^s F_{2s(n-1)} = F_s L_{s(2n-1)} $, showing a novel connection between Fibonacci and Lucas numbers through polynomial evaluation.