A new generalized bivariate polynomials and some properties

N. Saba, A. Boussayoud, N. Harrouche, B. Aloui, M. Kerada

In this paper, we first define a new generalization for bivariate polynomials denoted by {Wn(x,y)}n≥0 and then we obtain the Binet's formula to find the nth general term of these generalized bivariate polynomials. Also, the generating function and explicit formula are presented and proved. As an illustration, by considering the sequence {Wn(x,y)}n≥0, we give Binet's formulas, explicit formulas and generating functions of several special bivariate polynomials including bivariate Fibonacci, bivariate Mersenne Lucas polynomials, etc. After that, by using the symmetric functions we give some new generating functions for the products of these bivariate polynomials. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 18(3) (2025), pp. 11-39