Polynomial sequences: elementary basic methods and application hints. A survey

The present paper contains a small survey on the principal and elementary methods for a systematic study of general polynomial sequences. For the set of polynomial sequences the algebraic structure of a group is given. Matricial forms, recurrence relations and conjugate sequences of polynomials are...

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Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2019-10, Vol.113 (4), p.3829-3862
Main Authors: Costabile, Francesco Aldo, Gualtieri, Maria Italia, Napoli, Anna
Format: Article
Language:English
Subjects:
Applications of Mathematics
Differential equations
Fibonacci numbers
Interpolation
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Polynomials
Sequences
Survey
Theoretical
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Summary:The present paper contains a small survey on the principal and elementary methods for a systematic study of general polynomial sequences. For the set of polynomial sequences the algebraic structure of a group is given. Matricial forms, recurrence relations and conjugate sequences of polynomials are examined. For every element of a polynomial sequence determinant forms are determined by suitable Hessenberg matrices. Generating functions and derivation matrix are derived. Then, associated differential operator and Sheffer classification are considered. As an application, a general interpolation problem is hinted at. For a real valued function, the generalized Taylor polynomial and series are given. As an illustrative example, the shifted (with respect to the degree) Fibonacci polynomial sequence is considered.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-019-00682-9