[Paper Review] Star of David and other patterns in the Hosoya-like polynomials triangles

Rigoberto Flórez, Robinson A. Higuita+1 more|arXiv (Cornell University)|Jun 13, 2017

Advanced Mathematical Theories and Applications6 references3 citations

TL;DR

This paper generalizes Hosoya's numerical triangle to polynomial sequences using generalized Fibonacci polynomials (GFP), constructing Hosoya-like polynomial triangles where entries are products of GFP. It establishes the star of David property—equal GCDs of alternating vertices in a hexagonal pattern—for most GFP families, provides geometric proofs of Cassini’s, Catalan’s, and Johnson’s identities via rectangle and zigzag patterns, and classifies which triangles satisfy the property, with applications to known sequences like Fibonacci, Pell, and Chebyshev polynomials at x=1.

ABSTRACT

In this paper we first generalize the numerical recurrence relation given by Hosoya to polynomials. Using this generalization we construct a Hosoya-like triangle for polynomials, where its entries are products of generalized Fibonacci polynomials (GFP). Examples of GFP are: Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials and other familiar sequences of polynomials. For every choice of a GFP we obtain a triangular array of polynomials. In this paper we extend the star of David property, also called the Hoggatt-Hansell identity, to this type of triangles. We also establish the star of David property in the gibonomial triangle. In addition, we study other geometric patterns in these triangles and as a consequence we give geometric interpretations for the Cassini's identity, Catalan's identity, and other identities for Fibonacci polynomials.

Motivation & Objective

• To extend the star of David property (Hoggatt-Hansell identity) from numerical Hosoya triangles to polynomial triangles based on generalized Fibonacci polynomials (GFP).
• To provide geometric interpretations and proofs of classical identities like Cassini’s, Catalan’s, and Johnson’s for GFP using structural patterns in the triangle.
• To classify which families of GFP generate Hosoya polynomial triangles that satisfy the star of David GCD property.
• To explore numerical instances of these triangles by evaluating GFP at x=1, linking them to known integer sequences in the OEIS.
• To identify and analyze GCD patterns in polynomial coefficients, such as for Fermat and Pell polynomials, linking them to known integer sequences.

Proposed method

• Define generalized Fibonacci polynomials (GFP) via a second-order recurrence: $ G_n(x) = d(x)G_{n-1}(x) + g(x)G_{n-2}(x) $, with initial polynomials $ p_0(x), p_1(x) $, and $ \gcd(d(x),g(x))=1 $.
• Construct Hosoya-like polynomial triangles by placing entries $ H_{r,k} = G_k(x) G_{r-k}(x) $ in a triangular array.
• Use the Binet formula for GFP: $ G_n(x) = t_1 a^n(x) + t_2 b^n(x) $, where $ a(x), b(x) $ are roots of $ z^2 - d(x)z - g(x) = 0 $, to analyze algebraic structure.
• Apply geometric patterns—hexagons (star of David), rectangles, and zigzags—within the triangle to derive identities and GCD properties.
• Prove the star of David GCD property by analyzing GCDs of vertices in a hexagon and showing they are equal for most GFP families.
• Verify results numerically by evaluating entries at $ x=1 $, connecting the polynomial triangles to known integer sequences in the OEIS.

Experimental results

Research questions

• RQ1 Does the star of David property—equal GCDs of alternating vertices in a hexagon—hold in Hosoya polynomial triangles constructed from generalized Fibonacci polynomials?
• RQ2 Can classical identities like Cassini’s, Catalan’s, and Johnson’s be given geometric interpretations via rectangle and zigzag patterns in the Hosoya polynomial triangle?
• RQ3 For which families of generalized Fibonacci polynomials does the star of David GCD property hold, and what is the complete classification of such families?
• RQ4 What are the GCD patterns in the coefficients of specific GFPs like Fermat, Pell, and Chebyshev polynomials, and how do they relate to known integer sequences?
• RQ5 How do the polynomial triangles reduce to known numerical triangles (e.g., classic Hosoya triangle) when evaluated at $ x=1 $?

Key findings

• The star of David GCD property holds for most families of generalized Fibonacci polynomials in the Hosoya polynomial triangle, with a complete classification of the 14 distinct sub-families from Table 1.
• For the gibonomial triangle, the star of David property is established, extending earlier results on numerical triangles.
• Geometric proofs of Cassini’s, Catalan’s, and Johnson’s identities are derived using the rectangle and zigzag patterns in the Hosoya polynomial triangle.
• When evaluated at $ x=1 $, the Hosoya polynomial triangle entries yield known integer sequences in the OEIS, such as A058071 (Fibonacci), A284127 (Pell), and A143088 (Fermat).
• The GCD of coefficients of the $ n $-th Fermat polynomial $ \Phi_n(x) $ is $ 3^{a_n} $, where $ a_n $ is the $ n $-th term of OEIS sequence A168570.
• The GCD of coefficients of the $ 2n $-th Pell polynomial $ P_{2n}(x) $ is $ 2^{a_n} $, where $ a_n $ is the $ n $-th term of OEIS sequence A001511.