Generating Functions for Lacunary Legendre and Legendre-like Polynomials

On a generalization of the generating function for Gegenbauer polynomials

A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of several previously derived formulae such as Heine's formula and Heine's reciprocal square-root identity. We also show how this expansion can be used to compute hyperspherical harmonic expansions for power-law fundamental solutions of the polyharmonic equation.

On conformable fractional Legendre polynomials and their convergence properties with applications

The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide some of their main properties and convergence results. Subsequently, some pure recurrence and differential recurrence relations, Laplace's first integral formula and orthogonal properties are then developed for CFLPs. We append our study with presenting…

Algebraic Generating Functions for Gegenbauer Polynomials

It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments, if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given, which come from recently derived expressions for associated Legendre functions with octahedral or tetrahedral monodromy. It is also shown that if the Gegenbauer parameter is restricted as stated, the Poisson kernel for the Gegenbauer polynomials can be expressed in terms of complete elliptic integrals. An example is given.

Ferrers functions of arbitrary degree and order and related functions

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for Ferrers functions of the argument tanh(a+b) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for …

Complementary Polynomials From Rodrigues' Representations For Confluent And Hypergeometric Functions And More

Complementary polynomials of Legendre polynomials are briefly presented, as well as those for the confluent and hypergeometric functions, relativistic Hermite polynomials and corresponding new pre-Laguerre polynomials. The generating functions are all given in closed form and are much simpler than the standard ones. Some are simply polynomials in two variables. New recursions and addition formulas are derived.

Chenevier正交多项式的正性与渐近性

Positivity and Asymptotics for Chenevier's Orthogonal Polynomials

本文证明了Chenevier猜想的临界向量严格正性，通过Verblunsky系数符号和para-正交变换实现，并给出渐近分布与Legendre扰动的一阶变分公式。

帕斯卡矩阵、椭圆曲线上的点计数与普罗拉特球面函数

Pascal's Matrix, Point Counting on Elliptic Curves and Prolate Spheroidal Functions

研究帕斯卡矩阵的特征向量与普罗拉特球面函数的关系，通过椭圆曲线点计数验证生成函数的性质。