Differential Equations for Symmetric Generalized Ultraspherical Polynomials

Differential Equations for Symmetric Generalized Ultraspherical Polynomials

We look for differential equations satisfied by the generalized Jacobi polynomials { P n (x) }∞ n=0 which are orthogonal on the interval [−1, 1] with respect to the weight function Γ(α+ β + 2) 2α+β+1Γ(α+ 1)Γ(β + 1) (1− x)(1 + x) +Mδ(x+ 1) +Nδ(x− 1), where α > −1, β > −1, M ≥ 0 and N ≥ 0. In the special case that β = α and N = M we find all differential equations of the form ∞ ∑ i=0 ci(x)y (x) =...

Fractional differential equations with Legendre polynomials

Characterizations of Generalized Hermite and Sieved Ultraspherical Polynomials

A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the measure |x|γ(1 − x2)1/2dx is derived which is based on a “reversing property” of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalizati...

Ultraspherical Type Generating Functions for Orthogonal Polynomials

We characterize the probability distributions of finite all order moments having generating functions for orthogonal polynomials of ultraspherical type. 1. Motivation: Meixner families There is a one to one correspondance between probability distributions on the real line and polynomials of a one variable satisfying a three-terms recurrence relation subject to some positive conditions ([9]). Th...

Ultraspherical Type Generating Functions for Orthogonal Polynomials

We characterize, up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. 1. Motivation: Meixner families There is a one to one correspondance between probability distributions on the real line and polynomials of a one variable satisfying a three-terms recurrence relation subject to some positivity condit...