Pascal Matrix Representation of Evolution of Polynomials

Abstract

Pascal matrix is an adjoint operator of the differential operator of translation. This feature of the Pascal matrix is used in order to construct evolution equations for coefficients of polynomials induced by shifts of the roots. Under certain initial data solutions of the evolution equations are given by sequences of the Appell polynomials. The Pascal matrix is applied to calculate coefficients of the invariant polynomial and to construct a new algorithm of deflation.

The k-Tribonacci Matrix and the Pascal Matrix

“…Brawer and Pirovino [6] provide a form of matrix representation of the Pascal's triangle and discusses the algebraic form of the Pascal's matrix. Pascal matrix is an adjoint operator of the diferensial operator of translation [7].…”

Matrix approach to hypercomplex Appell polynomials

“…We mention, for instance, the approach developed in [30], which makes use of the generalized Pascal functional matrices and the characterization proposed in [14] which is based on a determinantal definition. As previously quoted, the authors introduced in [1] a matrix approach in the real case which has already been applied in the context of "image synthesis" [27], and in connection with "evolution equations" for the construction of a new algorithm of deflation [29].…”