Nonnegative and Strictly Positive Linearization of Jacobi and Generalized Chebyshev Polynomials

Abstract

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))_{nin mathbb {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),ldots ,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 (alpha ,beta )in (-1,infty )^2 for which the corresponding Jacobi polynomials (R_n^{(alpha ,beta )}(x))_{nin mathbb {N}_0}, normalized by R_n^{(alpha ,beta )}(1)equiv 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^{(alpha ,beta )}(x))_{nin mathbb {N}_0}, which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure (1-x^2)^{alpha }|x|^{2beta +1}chi _{(-1,1)}(x),mathrm {d}x. In this paper, we give the solution and show that (T_n^{(alpha ,beta )}(x))_{nin mathbb {N}_0} satisfies nonnegative linearization of products if and only if (alpha ,beta )in 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.