Exceptional Legendre Polynomials and Confluent Darboux Transformations

Painlevé III（$D_7$）方程代数解的合流 Darboux 变换与 Wronski 行列式

Confluent Darboux transformations and Wronskians for algebraic solutions of the Painlevé III ($D_7$) equation

本文利用 Schrödinger 算子的合流 Darboux 变换构造 Painlevé 方程代数解的 Wronskian 公式，并给出对应 $D_7$ 型 Painlevé III 方程代数解的 Ohyama 多项式新表示。

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.

Exceptional Gegenbauer polynomials via isospectral deformation

We show a method to construct isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, and it allows to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of continuous parameters. We propose to call these new orthogonal polynomial systems \emph{exceptional polynomials of the second kind}. We illustrate this construction by describing the class of exceptional Gegenbauer polynomials of the second kind.

A correlation function for the classical orthogonal polynomials

A correlation function of the classical orthogonal polynomials is defined and determined. The correlation function obeys a second order difference equation in two variables. The correlation function for the Gegenbauer, Chebyshev and Legendre polynomials can be written as a 4F3 hypergeometric function. For the Jacobi polynomials the result is an F2 Appell function. For the Generalized Laguerre polynomials the result is a confluent hypergeometric function and for the Hermite polynomials there rests only a single term.

Asymptotics of orthogonal polynomials for a weight with a jump on [-1,1]

We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^α(1+x)^β Ξ_{c}(x), \quad α, β>-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $Ξ_{c}$ is a step-like function: $Ξ_{c}(x)=1$ for $x\in [-1, 0)$ and $Ξ_{c}(x)=c^2$, $c>0$, for $x\in [0, 1]$. We obtain strong uniform asymptotics of the monic orthogonal polynomials in $\mathbb{C}$, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as $n\to \infty$. In particular, we p…