发表机构
University of Oregon（俄勒冈大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了作为二阶或四阶线性微分算子本征函数的多变量包裹乘积正交多项式，补充了高维相关结果并给出三维情形的完整算子列表。
AI 中文摘要
我们研究了d = d₁+d₂个变量的正交多项式，其对应于包裹乘积权函数W(x,y) = W₁(x/ρ(y))W₂(y)，其中(x,y) ∈ ℝ^d₁ × ℝ^d₂，ρ要么是线性的，要么是非负二次多项式的平方根；我们还确定了所有这类多项式中，作为二阶线性微分算子本征函数的那些多项式。对于d=2，已知在仿射变换下主要有五族这类多项式，它们是单变量经典正交多项式的乘积或包裹乘积，这五族在高维中都有对应，但在三维及更高维中尚无相关刻画。本研究探索了可行的包裹乘积族，找到了d = d₁+d₂个变量的两族新正交多项式对应的显式二阶微分算子，当d₁>1或d₂>1时，这两族多项式此前未被研究过；特别地，本研究给出了d=3时所有包裹乘积正交多项式中这类算子的完整列表。该列表还包含四族作为四阶微分算子本征函数的多项式，而它们没有对应的二阶算子。此外，我们还研究了包裹二次曲面上的正交多项式，这些多项式是曲面上二阶微分算子的本征值。
英文摘要
We study orthogonal polynomials of $d = d_1+d_2$ variables with respect to a wrapped product weight function ${\bm W}({\bm x},{\bm y}) = W_1({\bm x}/ρ({\bm y})) W_2(\bm y)$ for $(\bm x, \bm y) \in \mathbb{R}^{d_1} \times \mathbb{R}^{d_2}$, where $ρ$ is either linear or the square root of a nonnegative quadratic polynomial, and identify all such polynomials that are eigenfunctions of a second-order linear differential operator. For $d =2$, it is known that there are primarily, up to affine transformations, five families of such polynomials, which are products or wrapped products of classical orthogonal polynomials of one variable; all five families have their counterparts in higher dimensions, but no characterization is known in dimension three or higher. Our study explores viable wrapped product families, finds explicit second-order differential operators for two new families of orthogonal polynomials in $d= d_1+d_2$ variables that have not been studied before if either $d_1>1$ or $d_2 > 1$, and provides, in particular, a complete list of such operators among all wrapped product orthogonal polynomials when $d = 3$. The list also includes four families that are eigenfunctions of a fourth-order differential operator, whereas no second-order operator is available. Moreover, orthogonal polynomials on the wrapped quadratic surfaces that are eigenvalues of a second-order differential operator on the surface are also studied.
