正交多项式的保实根微分算子表示
Real-Root Preserving Differential Operator Representations of Orthogonal Polynomials
摘要：
本文研究了$T[x^n]=P_n(x)$形式的线性变换，其中$T[x^n]=P_n(x)$是一个正交多项式系统。特别感兴趣的是了解这些运营商保持真正的根多项式。已知当$P_n(x)$是Hermite多项式或标准Laguerre多项式时，变换$T$具有该性质。还已知变换$T[x^n]=H_n^α(x)$具有微分算子表示$T[x^n]=e^{-\fracα{2}D^2}x^n$，其中$H_n^α(x)$是具有实参数$α$的$n$次广义Hermite多项式。本文的主要结果是证明了$\sum_{k=0}^\infty \frac{γ_k}{k!} D^k$形式的微分算子诱导一元正交多项式组的充要条件是$\sum_{k=0}^\infty \frac{γ_k}{k!} D^k=γ_0e^{-\frac{ α}{2}D^2-βD}$，其中$γ_0,α,β\in \mathbb{C}$和$α,γ_0 \neq 0$ .当$α\in \mathbb{R}$。我们还表示转换从标准的基础上的标准拉盖尔基础，$T[x^n]=L_n(x)$作为一个微分算子的形式$\sum_{k=0}^\infty \frac{p_k(x)}{k!} D^k$其中的$p_k$是多项式，一个身份，以前没有被证明。
英文摘要：
In this paper, we study linear transformations of the form $T[x^n]=P_n(x)$ where $\{P_n(x)\}$ is an orthogonal polynomial system. Of particular interest is understanding when these operators preserve real-rootedness in polynomials. It is known that when the $P_n(x)$ are the Hermite polynomials or standard Laguerre polynomials, the transformation $T$ has this property. It is also known that the transformation $T[x^n]=H_n^α(x)$, where $H_n^α(x)$ is the $n$th generalized Hermite Polynomial with real parameter $α$, has the differential operator representation $T[x^n]=e^{-\fracα{2}D^2}x^n$. The main result of this paper is to prove that a differential operator of the form $\sum_{k=0}^\infty \frac{γ_k}{k!} D^k$ induces a system of monic orthogonal polynomials if and only if $\sum_{k=0}^\infty \frac{γ_k}{k!} D^k=γ_0e^{-\frac{ α}{2}D^2-βD}$ where $γ_0,α,β\in \mathbb{C}$ and $α,γ_0 \neq 0$. This operator will produce a shifted set of generalized Hermite polynomials when $α\in \mathbb{R}$. We also express the transformation from the standard basis to the standard Laguerre basis, $T[x^n]=L_n(x)$ as a differential operator of the form $\sum_{k=0}^\infty \frac{p_k(x)}{k!} D^k$ where the $p_k$ are polynomials, an identity that has not previously been shown.
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