Finding differential equations for symmetric generalized ultraspherical polynomials by using inversion methods

Differential equations for symmetric generalized ultraspherical polynomials

We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to the classical weight function for the Jacobi polynomials together with point masses at both endpoints. In the special symmetric case that both parameters are equal and also the two point masses are equal we find all differential equations of spectral type satisfied by these symmetric generalized ultraspherical polynomials. We show that if the point masses are positive only for nonnegative integer values of the parameter there exists exactly one differen…

Ultraspherical moments on a set of disjoint intervals

Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments.

关于超球多项式的Turán不等式

On Turán inequality for ultraspherical polynomials

本文证明了归一化超球多项式满足带权重因子 $|x|^θ$ 的更强 Turán 不等式，并将类似推广结果扩展到由三项递推定义的对称正交多项式。

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultra…

雅可比多项式与超球多项式的零点

Zeros of Jacobi and Ultraspherical polynomials

本文围绕雅可比与超球多项式零点交错问题，研究不同参数、次数组合下的交错性质，证明多类情形下仅部分交错而非完全交错，并通过数值例子验证结论的最优性。

源在曲线上时受限横向射线变换的反演

Inversion of a restricted transverse ray transform with sources on a curve

本文研究源在满足 Kirillov-Tuy 条件的固定曲线上时的受限横向射线变换，提出反演算法，通过重建加权 Radon 变换显式恢复向量场和对称 m-张量场。