Search

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence (P n (x)) n ∈ N 0 satisfies nonnegative linearization of products, i.e., the product of any two P m (x) , P n (x) is a conical combination of the polynomials P | m - n | (x) , ... , P m + n (x) . Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able to determine the set V of all pairs (α , β) ∈ (- 1 , ∞) 2 for which the corresponding Jacobi polynomials (R n (α , β) (x)) n ∈ N 0 , normalized by R n (α , β) (1) ≡ 1 , satisfy nonnegative linearization of products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked to solve the analogous problem for the generalized Chebyshev polynomials (T n (α , β) (x)) n ∈ N 0 , which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure (1 - x 2) α | x | 2 β + 1 χ (- 1 , 1) (x) d x . In this paper, we give the solution and show that (T n (α , β) (x)) n ∈ N 0 satisfies nonnegative linearization of products if and only if (α , β) ∈ V , so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper's original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper's one. [ABSTRACT FROM AUTHOR]

In this contribution, we first present a new recursion relation fulfilled by the linearization coefficients of Bessel polynomials (LCBPs), which is different than the one presented by Berg and Vignat in 2008. We will explain why this new recursion formula is as important as Berg and Vignat's. We introduce the matrix linearization coefficients of Bessel polynomials (MLCBPs), and we present some new results and some conjectures on these matrices. Second, we present the inverse of the connection coefficients with an application involving the modified Bessel function of the second kind. Finally, we introduce the inverse of the matrix of the linearization coefficients of the Bessel polynomials (IMLCBPs), and we present some open problems related to these IMLCBPs. [ABSTRACT FROM AUTHOR]

By using the three-term recurrence relation for orthogonal polynomials, we produce a collection of two-dimensional contiguous relations for certain generalized hypergeometric functions. These generalized hypergeometric functions arise through linearization coefficients for some classical orthogonal polynomials in the Askey-scheme, namely Gegenbauer (ultraspherical), Hermite, Jacobi and Laguerre polynomials. [ABSTRACT FROM AUTHOR]

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm. [ABSTRACT FROM AUTHOR]