Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6 |

Department of Mathematics and Computer Science

The Citadel

Charleston, SC 29409

USA

Robinson A. Higuita

Instituto de Matemáticas

Universidad de Antioquia Medellín

Colombia

Antara Mukherjee

Department of Mathematics and Computer Science

The Citadel

Charleston, SC 29409

USA

**Abstract:**

We define two types of second-order polynomial sequences. A sequence is
of Fibonacci-type (Lucas-type) if its Binet formula is similar in
structure to the Binet formula for the Fibonacci (Lucas) numbers.
Familiar examples are Fibonacci polynomials, Chebyshev polynomials,
Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat
polynomials, Jacobsthal polynomials, Vieta polynomials and other known
sequences of polynomials.

We generalize the numerical recurrence relation given by Hosoya to polynomials by constructing a Hosoya triangle for polynomials where each entry is either a product of two polynomials of Fibonacci-type or a product of two polynomials of Lucas-type. For every such choice of polynomial sequence we obtain a triangular array of polynomials. In this paper we extend the star of David property, also called the Hoggatt-Hansell identity, to these types of triangles. In addition, we study other geometric patterns in these triangles and as a consequence we obtain geometric interpretations for the Cassini identity, the Catalan identity, and other identities for Fibonacci polynomials.

(Concerned with sequences A001511 A007814 A058071 A141678 A143088 A168570 A284115 A284126 A284127 A284128 A284129 A284130 A284131 A284413.)

Received May 31 2017; revised versions received March 5 2018; March 20 2018; April 15 2018.
Published in *Journal of Integer Sequences*, May 8 2018.

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