Nonnegative and Strictly Positive Linearization of Jacobi and Generalized Chebyshev Polynomials

Abstract In the theory of orthogonal polynomials, as well in its intersection with harmonic analysis, it is an important problem to decide whether a given polynomial sequence $$(P_n(x))_{n\in \mathbb {N}_0}$$ ( P n x ) ∈ N 0 satisfies nonnegative linearization products, i.e., product any two $$P_m(x),P_n(x)$$ m , conical combination polynomials $$P_{|m-n|}(x),\ldots ,P_{m+n}(x)$$ | - … + . Since coefficients arising expansions are often cumbersome structure or not explicitly available, such considerations generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able determine set V all pairs $$(\alpha ,\beta )\in (-1,\infty )^2$$ α β 1 ∞ 2 for which corresponding Jacobi $$(R_n^{(\alpha )}(x))_{n\in R , normalized by $$R_n^{(\alpha )}(1)\equiv 1$$ ≡ satisfy products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked solve analogous generalized Chebyshev $$(T_n^{(\alpha T quadratic transformations and w.r.t. measure $$(1-x^2)^{\alpha }|x|^{2\beta +1}\chi _{(-1,1)}(x)\,\mathrm {d}x$$ χ d this paper, we give solution show that products if only V$$ V so share property polynomials. Moreover, reconsider themselves, simplify Gasper’s original proof characterize strict positivity coefficients. Our results can also be regarded sharpenings one.