## Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

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## Abstract:

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is studied and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.## References

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## Additional Information

**Yuan Xu**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- MR Author ID: 227532
- Email: yuan@uoregon.edu
- Received by editor(s): January 28, 2020
- Received by editor(s) in revised form: September 17, 2020
- Published electronically: February 2, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 3603-3657 - MSC (2020): Primary 42C05, 42C10, 33C50, 35P10
- DOI: https://doi.org/10.1090/tran/8312
- MathSciNet review: 4237958