A basic class of symmetric orthogonal polynomials using the extended Sturm–Liouville theorem for symmetric functions

Abstract

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞, ∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.

Some applications of a hypergeometric identity

“…A decomposition formula for classical hypergeometric orthogonal polynomials: there are ten sequences of hypergeometric polynomials [7,[12][13][14] that are orthogonal with respect to the Pearson distribution family Five of them are infinitely orthogonal with respect to special cases of the two above-mentioned weight functions and five other ones are finitely orthogonal [12][13][14] which are limited to some parametric constraints. The following Table 1 shows their main characteristics.…”

A basic class of symmetric orthogonal functions using the extended Sturm–Liouville theorem for symmetric functions

“…The other standard properties of orthogonal functions (23.1) such as generating function, integral representation, hypergeometric representation and so on can directly be obtained by using (25) and referring to [7]. For instance, we proved in [7] that…”

“…The generalized Hermite polynomials were first introduced by Szego who presented a second order differential equation for them [10,Problem 25] almost as the same form as indicated in [7]. These polynomials can be characterized by a direct relationship between them and Laguerre orthogonal polynomials [9].…”

“…First, let us recall that by using Favard theorem for finite orthogonal polynomials, two finite sequences of symmetric orthogonal polynomials were introduced in [7]. See also [6] in this regard.…”

“…For instance, by this theorem a basic class of symmetric orthogonal polynomials (BCSOP) is introduced in [7]. Since the main properties of BCSOP and also the extended Sturm-Liouville theorem for symmetric functions are needed in this article, we restate them here.…”