Polynomials Defined by Three-Term Recursion Relations and Satisfying a Second Recursion Relation: Connection with Discrete Integrability, Remarkable (Often Diophantine) Factorizations

DOI

10.1142/S1402925111001416How to use a DOI?

Keywords

Discrete integrability; recursion relations; orthogonal polynomials; Diophantine factori-zations; Askey polynomial classification

Abstract

In this paper (as in previous ones) we identify and investigate polynomials pn(ν)(x) featuring at least one additional parameter ν besides their argument x and the integer n identifying their degree. They are orthogonal (provided the parameters they generally feature fit into appropriate ranges) inasmuch as they are defined via standard three-term linear recursion relations; and they are interesting inasmuch as they obey a second linear recursion relation involving shifts of the parameter ν and of their degree n, and as a consequence, for special values of the parameter ν, also remarkable factorizations, often having a Diophantine connotation. The main focus of this paper is to relate our previous machinery to the standard approach to discrete integrability, and to identify classes of polynomials featuring these remarkable properties.

Copyright

© 2011 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - M. Bruschi
AU  - F. Calogero
AU  - R. Droghei
PY  - 2021
DA  - 2021/01/07
TI  - Polynomials Defined by Three-Term Recursion Relations and Satisfying a Second Recursion Relation: Connection with Discrete Integrability, Remarkable (Often Diophantine) Factorizations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 205
EP  - 243
VL  - 18
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925111001416
DO  - 10.1142/S1402925111001416
ID  - Bruschi2021
ER  -