Nonnegative and strictly positive linearization of Jacobi and generalized Chebyshev polynomials · arXivDesk
1812.05542Dec 13, 2018The second version puts more emphasis on strictly positive linearization of Jacobi polynomials. We reorganized the structure, added several references and corrected a few typos. We added a geometric interpretation of the set $V^{\prime}$ and some comments on its interior. We added a detailed comparison to Gasper's classical result. Title and abstract were changed. These are the main changes
Nonnegative and strictly positive linearization of Jacobi and generalized Chebyshev polynomials
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 (Pn(x))n∈N0 satisfies nonnegative linearization of products, i.e., the product of any two Pm(x),Pn(x)
Nearby in the stack
is a conical combination of the polynomials
P∣m−n∣(x),…,Pm+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
(α,β)∈(−1,∞)2
for which the corresponding Jacobi polynomials
(Rn(α,β)(x))n∈N0
, normalized by
Rn(α,β)(1)≡1
, satisfy nonnegative linearization of products. In 2005, R. Szwarc asked to solve the analogous problem for the generalized Chebyshev polynomials
(Tn(α,β)(x))n∈N0
, which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure
(1−x2)α∣x∣2β+1χ(−1,1)(x)dx
. In this paper, we give the solution and show that
(Tn(α,β)(x))n∈N0
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.