ABSTRACT

The Laguerre-Sobolev polynomials form an orthogonal polynomial system on the positive half-line with respect to the classical Laguerre measure with parameter $\alpha >-1$ and, in general, two point masses $N\ge 0$, S>0 at the origin involving functions and their first derivatives. Moreover we consider the Jacobi-Sobolev polynomials on the interval $[-1,1]$ with Jacobi parameters $\alpha ,\text{β}>-1$ and one Sobolev point mass S>0 at the right end-point of the domain. For $\alpha \in {\mathbb{N}}_0$, both polynomial systems are known to arise as eigenfunctions of certain spectral differential operators of finite order $2\alpha +8$, provided that N = 0 in the Laguerre case. In this paper, new representations of the two differential operators are established which consist of a number of elementary components reflecting the influence of the parameters, appropriately. In particular, we show that the operators are symmetric with respect to the respective Sobolev inner products and so recover the orthogonality of the eigenfunctions.

摘要

Laguerre-Sobolev 多项式相对于带参数的经典 Laguerre 测度在正半线上形成正交多项式系统 $\alpha >-1$ 并且，一般来说，两点质量 $N\ge 0$, S > 0 在涉及函数及其一阶导数的原点。此外，我们考虑区间上的 Jacobi-Sobolev 多项式$[-1,1]$ 使用雅可比参数 $\alpha ,\text{β}>-1$并且在域的右端点有一个 Sobolev 点质量S > 0。为了$\alpha \in {\mathbb{N}}_0$，已知这两个多项式系统都是作为某些有限阶谱微分算子的特征函数出现的 $2\alpha +8$，前提是 在 Laguerre 情况下N = 0。在本文中，建立了两个微分算子的新表示，它们由许多基本分量组成，适当地反映了参数的影响。特别是，我们表明算子关于各自的 Sobolev 内积是对称的，因此恢复了特征函数的正交性。