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 \((P_n(x))_{n\in \mathbb {N}_0}\) 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 \((R_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}\), 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 \((T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}\), which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure \((1-x^2)^{\alpha }|x|^{2\beta +1}\chi _{(-1,1)}(x)\,\mathrm {d}x\). In this paper, we give the solution and show that \((T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}\) 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.

在正交多项式理论中，以及在其与调和分析的交集中，确定给定的正交多项式序列\((P_n(x))_{n\in \mathbb {N}_0} \)满足乘积的非负线性化，即任意两个\(P_m(x),P_n(x)\)的乘积是多项式\(P_{|mn|}(x),\ldots的圆锥组合， P_{m+n}(x)\)。由于出现展开式中的系数通常具有繁琐的结构或不明确可用，因此此类考虑通常非常重要。Gasper (Can J Math 22:582–593, 1970) 能够确定所有对的集合V \((\alpha ,\beta )\in (-1,\infty )^2\)其中对应的雅可比多项式\((R_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}\)，通过\(R_n^{(\alpha , \beta )}(1)\equiv 1\)，满足乘积的非负线性化。Szwarc（因采尔讲座正交多项式，进阶论规格，功能该正交多项式，第2卷，新科学，公布，哈帕克，NY第103-139，2005）要求解决类似问题的广义切比雪夫多项式\（ (T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}\)，它们是雅可比多项式的二次变换和正交 wrt 测度\((1- x^2)^{\alpha }|x|^{2\beta +1}\chi _{(-1,1)}(x)\,\mathrm {d}x\)。在本文中，我们给出了解决方案并证明\((T_n^{(\alpha ,\beta )}(x))_{n\in \mathbb {N}_0}\)当且仅当\((\alpha ,\beta ) \in V\)，因此广义切比雪夫多项式与雅可比多项式共享此属性。此外，我们重新考虑 Jacobi 多项式本身，简化 Gasper 的原始证明并刻画线性化系数的严格正性。我们的结果也可以看作是加斯帕的结果的锐化。