Abstract: In this work, a Sobolev inner product on the unit ball of ?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.

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