Polynomial sequences generated by infinite Hessenberg matrices

Luis Verde-Star. "Polynomial sequences generated by infinite Hessenberg matrices." Special Matrices 5.1 (2017): 64-72. <http://eudml.org/doc/288071>.

@article{LuisVerde2017,
abstract = {We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.},
author = {Luis Verde-Star},
journal = {Special Matrices},
keywords = {Polynomial sequences of interpolatory type; infinite Toeplitz matrices; infinite Hessenberg matrices; Toeplitz companion matrices; polynomial sequences of interpolatory type; orthogonal polynomial sequences; recurrence relations; matrix similarity; generating functions},
language = {eng},
number = {1},
pages = {64-72},
title = {Polynomial sequences generated by infinite Hessenberg matrices},
url = {http://eudml.org/doc/288071},
volume = {5},
year = {2017},
}

TY - JOUR
AU - Luis Verde-Star
TI - Polynomial sequences generated by infinite Hessenberg matrices
JO - Special Matrices
PY - 2017
VL - 5
IS - 1
SP - 64
EP - 72
AB - We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.
LA - eng
KW - Polynomial sequences of interpolatory type; infinite Toeplitz matrices; infinite Hessenberg matrices; Toeplitz companion matrices; polynomial sequences of interpolatory type; orthogonal polynomial sequences; recurrence relations; matrix similarity; generating functions
UR - http://eudml.org/doc/288071
ER -