On an orthogonal polynomial set

A Set of Orthogonal Polynomials Induced by a Given Orthogonal Polynomial

Given an integer n ~ 1, and the orthogonal polynomial1rn (· ; du) of degree n relative to some positive measure du, the polynomial system "induced" by 1rn is the system of orthogonal polynomials {7rk,n} corresponding to the modified measure dUn = 1r~du. Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials 7rk,n from the re...

An Orthogonal Property of the Hypergeometric Polynomial.

1. Mittag-Leffler's polynomial gn(z) has the orthogonal property fgm(-ix)gn(ix)dx/xsh(x) = m $ n m > 0-co =-2/n m=n n>O. (1.1) This is readily obtained by inverting the integral representation' co gn(ix) = (1/ir) sin (irx)f exu (tanh '/2u)ndu/sh u n > 1 (1.2)-co0 by means of Fourier's inversion formula. The resulting equation co cosech u (tanh 1/2u)n = 1/2 e-iux cosech (irx)gn(ix)dx (1.3)-OD th...

On orthogonal polynomials obtained via polynomial mappings

Let (pn)n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k ≥ 2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS (qn)n and two polynomials πk and θm , with degrees k and m (resp.), with 0 ≤ m ≤ k − 1, such that pnk+m(x) = θm(x)qn(πk(x)) (n = 0, 1, 2, . . .). In...

Orthogonal polynomial expansions on sparse grids

We study the orthogonal polynomial expansion on sparse grids for a function of d variables in a weighted L space. A fast algorithm is developed to compute the orthogonal polynomial expansion by combining the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space, and a fast algorithm of computing hierarc...

On Bochner-krall Orthogonal Polynomial Systems