\(d\)-Gaussian Fibonacci, \(d\)-Gaussian Lucas polynomials, and their matrix representations (Q6083228)

The authors define the \(d\)-Gaussian Fibonacci polynomials \(GF_n(x)\) by \[ GF_{n+1}(x)=p_1(x)GF_n(x)+p_2(x)GF_{n-1}(x)+\ldots+p_{d+1}(x)GF_{n-d}(x) \] with \(GF_n(x)=0\) for \(n\le 0\) and \(GF_1(x)=p_1(x)+i\). Probably \(p_1(x),\ldots,p_{d+1}(x)\) are given real polynomials, and \(i\) is a given real number. They also define the \(d\)-Gaussian Lucas polynomials \(GL_n(x)\) by the above recursion with \(GL_n(x)=0\) for \(n\le 0\) and \(GL_1(x)=p_1(x)+2i\). The notions of Gaussian Fibonacci and Gaussian Lucas polynomials [\textit{E. Özkan} and \textit{M. Taştan}, Commun. Algebra 48, No. 3, 952--960 (2020; Zbl 1491.11022)] are so generalized. From the abstract: ``By using the Riordan method, we obtain the factorizations of the Pascal matrix, including polynomials. In addition, we define the infinite \(d\)-Gaussian Fibonacci polynomial matrix and the \(d\)-Gaussian Lucas polynomial matrix and present their inverses.''