The main goal of the paper is to introduce methods which compute Bézier curves faster than Casteljau’s method does. These methods are based on the spectral factorization of a n×n Bernstein matrix, B n(s) = PnGn(s)P −1 n , where Pn is the n×n lower triangular Pascal matrix. So we first calculate the exact optimum positive value t in order to transform Pn in a scaled Toeplitz matrix, which is a p...

In this paper, we derive a family of fast and stable algorithms for multiplying and inverting n × n Pascal matrices that run in O(n log n) time and are closely related to De Casteljau’s algorithm for Bézier curve evaluation. These algorithms use a recursive factorization of the triangular Pascal matrices and improve upon the cripplingly unstable O(n log n) fast Fourier transform-based algorithm...

In this paper, we prove the q -analogue of the fundamental theorem of Riordan arrays. In particular, by defining two new binary operations ∗q and ∗1/q , we obtain a q -analogue of the Riordan representation of the q -Pascal matrix. In addition, by aid of the q -Lagrange expansion formula we get q -Riordan representation for its inverse matrix.

In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. particular, focus our attention on matrix-inversion formulae involving these matrices. Furthermore, provide analytic so-called Pascal matrix of first kind, factorizations Euler matrix.