A Combinatorial Proof of a Result on Generalized Lucas Polynomials

Abstract

We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.

Keywords

• binomial theorem
• Fibonacci number
• Fibonomial coefficient
• Lucas number q-analogue
• Generalized Lucas Polynomials