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,… Click to show full 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.
               
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