LINEAR COMBINATIONS OF 2-ORTHOGONAL POLYNOMIALS: GENERATION AND DECOMPOSITION PROBLEMS

AHMED NASRI; AMMAR BOUKHEMIS; FRANCISCO MARCELLÁN ESPAÑOL

Title	LINEAR COMBINATIONS OF 2-ORTHOGONAL POLYNOMIALS: GENERATION AND DECOMPOSITION PROBLEMS
Publication Type	Journal Article
Year of Publication	2018
Authors	NASRI, AHMED, BOUKHEMIS, AMMAR, ESPAÑOL, FRANCISCOMARCELLÁ
Secondary Title	LINEAR COMBINATIONS OF 2-ORTHOGONAL POLYNOMIALS
Volume	22
Issue	1
Start Page	97
Pagination	24
Date Published	01/2018
Type of Work	scientific: mathematics
ISSN	1083-2564
AMS	33C45, 42C05
Abstract	In this work we are interested in the study of the $2$-orthogo\-nality of sequences of monic $2$-orthogonal polynomials $\left \{ P_{n}\right \} _{n\geq \text{ }\text{ }0}$ and $\left \{ Q_{n}\right \} _{n\geq 0}$ satisfying the relation $Q_{n+1}(x)=P_{n+1}(x)+\alpha _{n+1}P_{n}(x),$ $n\geq $ $0,$where $\alpha _{n},$ $n\geq $ $1,$ are nonzero complex numbers. The sequence $\left \{ Q_{n}\right \} _{n\geq 0}$ is said to be generated with $2$ terms of the sequence $\left \{ P_{n}\right \} _{n\geq 0}$ and the sequence $\left \{ P_{n}\right \} _{n\geq 0}$ is said to be a decomposition of the sequence $\left \{ Q_{n}\right \} _{n\geq 0}$ with $2$ terms. First, we give necessary and sufficient conditions for the $2$-orthogonality of the sequence $\left \{ Q_{n}\right \} _{n\geq 0}$ assuming the $2$-orthogonality of the sequence $\left \{ P_{n}\right \} _{n\geq 0}.$ Second, assuming the sequence $\left \{ Q_{n}\right \} _{n\geq 0}$ is $2$-orthogonal we get necessary and sufficient conditions for the existence of a sequence $\left \{ P_{n}\right \} _{n\geq 0}$ satisfying the above relation and such that it is $2$ orthogonal. Indeed, we characterize the $2$-orthogonality of these sequences in terms of the coefficients of the corresponding four term r\'{e}currence relations. Next, we study our problem as an inverse problem for $2$-monic orthogonal polynomials. Furthermore, the relation between the banded Hessenberg matrices associated with the multiplication operator in terms of \pagebreak the bases $\left \{ P_{n}\right \} _{n\geq 0}$ and $\left \{ Q_{n}\right \} _{n\geq 0}$ is analyzed. Finally, we give many examples of such related $2$-orthogonal polynomial sequences.
URL	http://www.acadsol.eu/en/articles/22/1/7.pdf
DOI	10.12732/caa.v22i1.7
Refereed Designation	Refereed
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