An Introductory Overview of Bessel Polynomials, the Generalized Bessel Polynomials and the q-Bessel Polynomials
Abstract
Named essentially after their close relationship with the modified Bessel function Kν(z) of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials yn(x) and the generalized Bessel polynomials yn(x;α,β) stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials yn(x) and the generalized Bessel polynomials yn(x;α,β) involving the asymmetric parameters α and β. Each of these polynomial systems, as well as their reversed forms θn(x) and θn(x;α,β), has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function rFs with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with rΦs which also involves r symmetric numerator parameters and s symmetric denominator parameters.
- Publication:
-
Symmetry
- Pub Date:
- March 2023
- DOI:
- 10.3390/sym15040822
- Bibcode:
- 2023Symm...15..822S
- Keywords:
-
- Bessel and generalized polynomials;
- basic (or quantum or q-) Bessel polynomials;
- Bessel functions and the modified Bessel functions;
- orthogonality properties;
- generating functions;
- polynomial expansions;
- asymptotic expansions and location of zeros;
- hypergeometric functions and hypergeometric polynomials;
- q-hypergeometric functions and q-hypergeometric polynomials;
- 33C20;
- 33C45;
- 33D15;
- 05A30;
- 11B65;
- 33C10;
- 33D50