On Generalized Bivariate (p,q)-Bernoulli–Fibonacci Polynomials and Generalized Bivariate (p,q)-Bernoulli–Lucas Polynomials

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, we define the generalized (p,q)-Bernoulli–Fibonacci and generalized (p,q)-Bernoulli–Lucas polynomials and numbers by using the (p,q)-Bernoulli numbers, unified (p,q)-Bernoulli polynomials, h(x)-Fibonacci polynomials, and h(x)-Lucas polynomials. We also introduce the generalized bivariate (p,q)-Bernoulli–Fibonacci and generalized bivariate (p,q)-Bernoulli–Lucas polynomials and numbers. Then, we derive some properties of these newly established polynomials and numbers by using their generating functions with their functional equations. Finally, we provide some families of bilinear and bilateral generating functions for the generalized bivariate (p,q)-Bernoulli–Fibonacci polynomials.

New Summation and Integral Representations for 2-Variable (p,q)-Hermite Polynomials

“…− p 2 yD p,q,x H 3,p,q (x, p 2 y) [3] p,q ! + p 4 y 2 D p,q,x H 1,p,q (x, p 2 y) [2] p,q ! = (px) 4 p,q…”

“…Recently, (p, q)-calculus got attention from the researchers of various fields of mathematics and physics. Presented and thoroughly investigated are the (p, q)-analogues of several conventional special functions, including the Hermite polynomials, Bernoulli polynomials, Euler polynomials, Beta function, Gamma function, generalized bivariate (p, q)-Bernoulli-Fibonacci polynomials and generalized bivariate (p, q)-Bernoulli-Lucas polynomials, family of (p, q)-hybrid polynomials and (p, q)-sine and (p, q)-cosine Fubini polynomials; for more details, we refer the readers to [1][2][3][4][5][6][7][8][9] and references therein.…”