Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 042, 26 pages      arXiv:1209.6047      https://doi.org/10.3842/SIGMA.2013.042

Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

Howard S. Cohl
Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA

Received November 29, 2012, in final form May 28, 2013; Published online June 05, 2013

Abstract
We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.

Key words: fundamental solutions; polyharmonic equation; Jacobi polynomials; Gegenbauer polynomials; Chebyshev polynomials; eigenfunction expansions; separation of variables; addition theorems.

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References

  1. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.
  2. Alonso Izquierdo A., Fuertes W.G., de la Torre Mayado M., Guilarte J.M., One-loop corrections to the mass of self-dual semi-local planar topological solitons, Nuclear Phys. B 797 (2008), 431-463, arXiv:0707.4592.
  3. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  4. Boyling J.B., Green's functions for polynomials in the Laplacian, Z. Angew. Math. Phys. 47 (1996), 485-492.
  5. Cohl H.S., Erratum: Developments in determining the gravitational potential using toroidal functions, Astronom. Nachr. 333 (2012), 784-785.
  6. Cohl H.S., Fourier and Gegenbauer expansions for fundamental solutions of the Laplacian and powers in Rd and Hd, Ph.D.  thesis, The University of Auckland, Auckland, New Zealand, 2010.
  7. Cohl H.S., Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation, arXiv:1202.1811.
  8. Cohl H.S., Dominici D.E., Generalized Heine's identity for complex Fourier series of binomials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), 333-345, arXiv:0912.0126.
  9. Cohl H.S., Kalnins E.G., Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, J. Phys. A: Math. Theor. 45 (2012), 145206, 32 pages, arXiv:1105.0386.
  10. Cohl H.S., Rau A.R.P., Tohline J.E., Browne D.A., Cazes J.E., Barnes E.I., Useful alternative to the multipole expansion of 1/r potentials, Phys. Rev. A 64 (2001), 052509, 5 pages, arXiv:1104.1499.
  11. Cohl H.S., Tohline J.E., Rau A.R.P., Srivastava H.M., Developments in determining the gravitational potential using toroidal functions, Astronom. Nachr. 321 (2000), 363-372.
  12. Cormen T.H., Leiserson C.E., Rivest R.L., Stein C., Introduction to algorithms, 2nd ed., MIT Press, Cambridge, MA, 2001.
  13. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, McGraw-Hill, New York - Toronto - London, 1953.
  14. Fano U., Rau A.R.P., Symmetries in quantum physics, Academic Press Inc., San Diego, CA, 1996.
  15. Gel'fand I.M., Shilov G.E., Generalized functions. Vol. I. Properties and operations, Academic Press, New York, 1964.
  16. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007.
  17. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  18. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. The n-dimensional sphere, J. Math. Phys. 40 (1999), 1549-1573.
  19. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and the separation of variables: interbase expansions, J. Phys. A: Math. Gen. 34 (2001), 521-554.
  20. Kil'dyushov M.S., Hyperspherical functions of the "tree" type in the n-body problem, Soviet J. Nuclear Phys. 15 (1972), 113-118.
  21. Koekoek J., Koekoek R., The Jacobi inversion formula, Complex Variables Theory Appl. 39 (1999), 1-18, math.CA/9908148.
  22. Lake M., Thomas S., Ward J., Non-topological cycloops, J. Cosmol. Astropart. Phys. 2010 (2010), no. 1, 026, 27 pages, arXiv:0911.3118.
  23. Lin C.D., Hyperspherical coordinate approach to atomic and other Coulombic three-body systems, Phys. Rep. 257 (1995), 1-83.
  24. Lin F., Yang Y., Energy splitting, substantial inequality, and minimization for the Faddeev and Skyrme models, Comm. Math. Phys. 269 (2007), 137-152.
  25. Magnus W., Oberhettinger F., Soni R.P., Formulas and theorems for the special functions of mathematical physics, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Bd. 52, Springer-Verlag, New York, 1966.
  26. Miller Jr. W., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass. - London - Amsterdam, 1977.
  27. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010.
  28. Schwartz L., Théorie des distributions. Tome I, Actualités Sci. Ind., no. 1091, Hermann & Cie., Paris, 1950.
  29. Sloane N.J.A., Plouffe S., The encyclopedia of integer sequences, Academic Press Inc., San Diego, CA, 1995.
  30. Stanley R.P., Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.
  31. Szegö G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
  32. Vilenkin N.Ja., Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, Amer. Math. Soc., Providence, R.I., 1968.
  33. Vilenkin N.Ja., Klimyk A.U., Representation of Lie groups and special functions. Vol. 2. Class I representations, special functions, and integral transforms, Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1993.
  34. Vilenkin N.Ja., Kuznetsov G.I., Smorodinskiĭ Ya.A., Eigenfunctions of the Laplace operator providing representations of the U(2), SU(2), SO(3), U(3) and SU(3) groups and the symbolic method, Soviet J. Nuclear Phys. 2 (1965), 645-652.
  35. Wen Z.Y., Avery J., Some properties of hyperspherical harmonics, J. Math. Phys. 26 (1985), 396-403.

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