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https://www.nist.gov/itl/math/acmd-seminar-binomial-and-logarithmic-gegenbauer-expansions-even-dimensional-polyharmonic

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# ACMD Seminar: Binomial and logarithmic Gegenbauer expansions for the  even-dimensional polyharmonic equation

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Howard Cohl*Applied and Computational Mathematics Division, ITL, NIST*

Monday, August 7, 2017, 15:00 - 16:00Building 101, Lecture Room CGaithersburg

Monday August 7, 2017, 13:00 - 14:00Room 1-4058Boulder

**Abstract:**On even-dimensional Euclidean space for integer powers of the Laplacian greater than or equal to the dimension divided by two, a fundamental solution of the polyharmonic equation has binomial and logarithmic behaviour. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit process applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. For the Gegenbauer logarithmic fundamental solution expansion, we use parameter derivatives applied to the Gegenbauer expansion of a power-law fundamental solution for the polyharmonic equation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we obtain addition theorems for the azimuthal Fourier coefficients in Vilenkin polyspherical standard geodesic polar coordinates and generalised Hopf coordinates.

**Bio:** Howard Cohl is a Mathematician in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology in Gaithersburg, Maryland.  He obtained a B.S. in Astronomy and Astrophysics from Indiana University, Bloomington, Indiana, M.S. and Ph.D. in Physics from Louisiana State University, Baton Rouge, Louisiana, and Ph.D. in Mathematics from the University of Auckland, Auckland, New Zealand.  Cohl is a member of the NIST Digital Library of Mathematical Functions and NIST Digital Repository of Mathematical Formulae teams and publishes papers on the properties of Special Functions and Orthogonal Polynomials (and their q-analgoues). He is particularly interested in fundamental solutions and their eigenfunctions expansions for linear partial differential equations on isotropic Riemannian manifolds.

**Note:** Visitors from outside NIST must contact Cathy Graham; (301) 975-3800; at least 24 hours in advance.

Mathematics and statistics

## Contacts

                -  Howard Cohl  howard.cohl@nist.gov      (202) 306-2705   

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