On (<i>p</i>,<i>q</i>)–Fibonacci and (<i>p</i>,<i>q</i>)–Lucas Polynomials Associated with Changhee Numbers and Their Properties

Abstract

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties have been studied in the literature with the help of generating functions and their functional equations. In this paper, using the (p,q)–Fibonacci polynomials, (p,q)–Lucas polynomials, and Changhee numbers, we define the (p,q)–Fibonacci–Changhee polynomials and (p,q)–Lucas–Changhee polynomials, respectively. We obtain some important identities and relations of these newly established polynomials by using their generating functions and functional equations. Then, we generalize the (p,q)–Fibonacci–Changhee polynomials and the (p,q)–Lucas–Changhee polynomials called generalized (p,q)–Fibonacci–Lucas–Changhee polynomials. We derive a determinantal representation for the generalized (p,q)–Fibonacci–Lucas–Changhee polynomials in terms of the special Hessenberg determinant. Finally, we give a new recurrent relation of the (p,q)–Fibonacci–Lucas–Changhee polynomials.

Keywords

• (<i>p</i>,<i>q</i>)–Fibonacci polynomials
• (<i>p</i>,<i>q</i>)–Lucas polynomials
• Changhee numbers
• generating function
• Hessenberg determinant