[Paper Review] Nonnegative and strictly positive linearization of Jacobi and generalized Chebyshev polynomials

Stefan Kahler|arXiv (Cornell University)|Dec 13, 2018

Mathematical functions and polynomials27 references4 citations

TL;DR

This paper solves the open problem of characterizing nonnegative linearization of products for generalized Chebyshev polynomials, proving they satisfy this property if and only if the parameters (α, β) lie in the same set V as for Jacobi polynomials. The authors simplify Gasper's original proof for Jacobi polynomials, characterize strict positivity of linearization coefficients, and provide corrected hypergeometric representations of these coefficients, offering a sharper and more elementary treatment of a classical problem in orthogonal polynomials and harmonic analysis.

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 $(P_n(x))_{n\in\mathbb{N}_0}$ 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. In 1970, G. Gasper 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 $R_n^{(\alpha,\beta)}(1)\equiv1$, satisfy nonnegative linearization of products. In 2005, R. Szwarc 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.

Motivation & Objective

• To determine the parameter range (α, β) ∈(−1, ∞)² for which generalized Chebyshev polynomials (T(α,β)n(x))n∈N0 satisfy nonnegative linearization of products.
• To provide a simplified and more elementary proof of Gasper's classical result on nonnegative linearization for Jacobi polynomials.
• To characterize strict positivity of the linearization coefficients gR(m, n; k) for Jacobi polynomials.
• To correct and refine the hypergeometric representations of the linearization coefficients gR(m, n; k) as presented in Rahman's work.
• To establish that the generalized Chebyshev polynomials inherit the nonnegative linearization property precisely when (α, β) ∈ V, mirroring the Jacobi case.

Proposed method

• The authors derive a new, simplified proof of Gasper's result on nonnegative linearization for Jacobi polynomials by analyzing the sign behavior of the linearization coefficient sequences.
• They introduce and analyze the function ι(m, n; k) to study the sign and oscillatory behavior of linearization coefficients, particularly focusing on extremal terms gR(m, n; |m−n|+2) and gR(m, n; m+n−2).
• A key technique involves rewriting the coefficient ι(m, m; j) in terms of a quartic polynomial χm(j), whose root structure determines the sign behavior and enables the derivation of necessary and sufficient conditions for nonnegativity.
• The paper corrects and re-expresses Rahman’s hypergeometric representations of the linearization coefficients gR(m, n; k), providing valid formulas for both even and odd j in the summation index.
• The authors use these corrected representations to prove nonnegativity and strict positivity of coefficients under specific parameter conditions.
• They establish the equivalence between nonnegative linearization for generalized Chebyshev polynomials and the same property for Jacobi polynomials by leveraging quadratic transformation identities.

Experimental results

Research questions

• RQ1 For which parameters (α, β) ∈(−1, ∞)² do the generalized Chebyshev polynomials (T(α,β)n(x))n∈N0 satisfy nonnegative linearization of products?
• RQ2 Can Gasper’s original proof for Jacobi polynomials be simplified and made more transparent using sign analysis of coefficient sequences?
• RQ3 What are the necessary and sufficient conditions for the strict positivity of the linearization coefficients gR(m, n; k) in the expansion of Pm(x)Pn(x)?
• RQ4 What corrections are needed in Rahman’s hypergeometric representations of the linearization coefficients gR(m, n; k), and how do they affect the nonnegativity and positivity analysis?
• RQ5 Is the nonnegative linearization property for generalized Chebyshev polynomials equivalent to that of the underlying Jacobi polynomials?

Key findings

• The generalized Chebyshev polynomials (T(α,β)n(x))n∈N0 satisfy nonnegative linearization of products if and only if (α, β) ∈ V, where V is the same parameter set identified by Gasper for Jacobi polynomials.
• The paper provides a simplified and more elementary proof of Gasper’s Theorem 1.1, avoiding the long computational steps of the original proof.
• Strict positivity of the linearization coefficients gR(m, n; k) is characterized: they are strictly positive if and only if α > −11/8 + √73/8 ≈ −0.15 and β > −1/2, with additional constraints on the parameters.
• The authors correct Rahman’s hypergeometric representations of the coefficients gR(m, n; k), providing valid formulas for both even and odd j in the summation index, and show that these representations confirm nonnegativity for (α, β) ∈ ∆ and strict positivity for (α, β) ∈ ∆◦.
• The corrected representations reveal that the coefficients are nonnegative when (α, β) ∈ ∆ and strictly positive when (α, β) ∈ ∆◦, with explicit conditions on the parameters.
• The study establishes that the nonnegative linearization property for generalized Chebyshev polynomials is equivalent to that of the corresponding Jacobi polynomials, with the same parameter set V.