In this paper, we compute terms of the matrix A ∞ (k) , which contains Fibonacci type numbers and polynomials, with the help of determinants and permanents of various Hessenberg matrices. In addition, we show that determinants of these Hessenberg matrices can be obtained by using combinations. The results that we obtain are important, since the matrix A ∞ (k) is a general form of Fibonacci type numbers and polynomials, such as k sequences of the generalized order-k Fibonacci and Pell numbers, generalized bivariate Fibonacci p-polynomials, bivariate Fibonacci and Pell p-polynomials, second kind Chebyshev polynomials and bivariate Jacobsthal polynomials, etc. 2010 AMS Mathematics subject classification. Primary 11B37, Secondary 15A15, 15A51.Keywords and phrases. Matrix A ∞ (k) , generalized Fibonacci polynomials, Hessenberg matrix.

No abstract

In this paper, we present a new approach for finding the inverse of some triangular Toeplitz matrices using the generalized Fibonacci polynomials and give a factorization of these matrices. We also give a new proof of Trudi's formula using our result.

No abstract

In the present article, we have discussed the (,)-numbers, the Rogers-Szegő polynomial and the (,)-Rogers-Szegő polynomial and have defined the (,)-matrices and the (,)-Rogers-Szegő matrices. We have presented some algebraic properties of these matrices and have proved them. In particular, we have obtained the factorization of these matrices, their inverse matrices, as well as the matrix representations of the (,)-numbers, the Rogers-Szegő polynomials and the (,)-Rogers-Szegő polynomials.