In this work, a Sobolev inner product on the unit ball of \(\mathbb {R}^{d}\) involving the outward normal derivative is considered. A basis of mutually orthogonal polynomials associated with this inner product is constructed in terms of spherical harmonics and a radial part obtained from a family of univariate polynomials orthogonal with respect to a Sobolev inner product. The properties of this family constitute our main subject of study. In particular, we deduce algebraic properties, connection formulas, and some asymptotic properties. Finally, we show some numerical tests to illustrate the behavior of the roots of these univariate non-standard orthogonal polynomials.

在这项工作中，考虑了包含向外法向导数的\（\ mathbb {R} ^ {d} \）单位球上的Sobolev内积。根据球谐函数和从相对于Sobolev内积正交的单变量多项式族中获得的径向部分，构造与该内积关联的相互正交的多项式的基础。这个家庭的财产构成了我们的主要研究主题。特别是，我们推导了代数性质，连接公式和一些渐近性质。最后，我们显示一些数值测试来说明这些单变量非标准正交多项式的根的行为。