‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

Abstract

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of $0,1,\dots ,N$, constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

Keywords:

• Finite orthogonal polynomials in the Askey scheme
• (basic) hypergeometric orthogonal polynomials
• Diophantine property
• multi-indexed orthogonal polynomials
• Darboux transformations

AMS Classifications:

• 33C45
• 33D45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by JSPS KAKENHI Grant Number JP19K03667.