Some Applications of the Sheffer A-Type 0 Orthogonal Polynomial Sequences

2012 ◽  
pp. 35-66
Author(s):  
Daniel Joseph Galiffa
Keyword(s):  
Orthogonal Polynomial ◽  
Polynomial Sequences

The Almost Orthogonal Polynomial Sequences

2010 ◽  
Author(s):  
M. S. Stanković ◽  
B. Danković ◽  
S. Marinković ◽  
P. M. Rajković ◽  
Michail D. Todorov ◽  
...  
Keyword(s):  
Orthogonal Polynomial ◽  
Polynomial Sequences

scholarly journals A characterization of the four Chebyshev orthogonal families

2005 ◽  
Vol 2005 (13) ◽  
pp. 2071-2079 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. Garcia-Amor
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Linear Combinations ◽  
Classical Orthogonal Polynomial ◽  
Polynomial Sequences ◽  
Fixed Length ◽  
Constant Coefficients ◽  
Fourth Kind

We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.

scholarly journals Polynomial sequences generated by infinite Hessenberg matrices

2017 ◽  
Vol 5 (1) ◽  
pp. 64-72 ◽  
Author(s):  
Luis Verde-Star
Keyword(s):  
Orthogonal Polynomial ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hessenberg Matrix ◽  
Polynomial Sequences ◽  
Lower Triangular ◽  
Companion Matrices ◽  
Matrix Similarity ◽  
Invertible Matrices ◽  
Construction Algorithms

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.

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