Generating functions of generalized Gaussian polynomials

Abstract

In this paper, we Örst deÖne new generalization for Gaussian polynomials fGWn (x)gn0 and then we obtain the Binetís formula to Önd the n th general term of generalized Gaussian polynomials fGWn (x)gn0 : After that, the ordinary generating functions and the explicit formulas of generalized Gaussian polynomials and (p; q)-Fibonacci-like numbers are obtained. Considering the sequence of generalized Gaussian polynomials, we give Binetís formulas, explicit formulas and ordinary generating functions of Gaussian Pell and Gaussian Pell Lucas polynomials, Gaussian Jacobsthal and Gaussian Jacobsthal Lucas polynomials, Gaussian Fibonacci and Gaussian Lucas polynomials. Also, we present and prove certain ordinary generating functions for the products of (p; q)-Fibonacci-like numbers with these Gaussian polynomials and the products of (p; q)-Fibonacci-like numbers with Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Jacobsthal numbers, Gaussian Jacobsthal Lucas numbers, Gaussian Pell numbers and Gaussian Pell Lucas numbers.