Nonnegative linearization of orthogonal polynomials
نویسندگان
چکیده
منابع مشابه
A Necessary and Sufficient Condition for Nonnegative Product Linearization of Orthogonal Polynomials
cos nθ cos mθ = 2 cos(n − m)θ + 2 cos(n + m)θ. Certain classical orthogonal polynomials admit explicit computation of the coefficients c(n,m, k). For example, they are known explicitly for the ultraspherical polynomials along with their q-analogs [8]. However, they are not available in a simple form for the nonsymmetric Jacobi polynomials (see [7]). The first general criterion for nonnegativity...
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The support of the orthogonality measure of so-called little q-Laguerre polynomials {ln(.; a|q)}n=0, 0 < q < 1, 0 < a < q−1, is given by Sq = {1, q, q, . . .} ∪ {0}. Based on a method of MÃlotkowski and Szwarc we deduce a parameter set which admits nonnegative linearization. We additionally use this result to prove that little q-Laguerre polynomials constitute a so-called Faber basis in C(Sq).
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The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials. In particular, we show that Viennot's combinatorial interpretations of the moments can be derived directly from their classical analytical exp...
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ABSTRACT. Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom...
متن کاملDiscriminants and nonnegative polynomials
For a semialgebraic set K in R, let Pd(K) = {f ∈ R[x]≤d : f(u) ≥ 0 ∀u ∈ K} be the cone of polynomials in x ∈ R of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irreducible hypersurface defined by the discriminant ∆(f) of f . When K = {x ∈ R : g1(x) = · · · = gm(x) = 0}...
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ژورنال
عنوان ژورنال: Colloquium Mathematicum
سال: 1996
ISSN: 0010-1354,1730-6302
DOI: 10.4064/cm-69-2-309-316