# American Institute of Mathematical Sciences

November  2010, 4(4): 639-647. doi: 10.3934/ipi.2010.4.639

## Special functions

 1 Department of Mathematics, Temple University, Philadelphia, PA 19122, United States

Received  March 2009 Published  September 2010

Special functions are functions that show up in several contexts. The most classical special functions are the monomials and the exponential functions. On the next level we find the hypergeometric functions, which appear in such varied contexts as partial differential equations, number theory, and group representations. The standard hypergeometric functions have power series which satisfy 2 term recursion relations. This leads to the usual expressions for the power series coefficients as quotionts of rational and factorial-like expressions. We have developed a "hierarchy" of special functions which satisfy higher order recursion relations. They generalize the classical Mathieu and Lamé functions. These classical functions satisfy 3 term recursion relations and our theory produces "Lamé - like" functions which satisfy recursions of any order.
Citation: Leon Ehrenpreis. Special functions. Inverse Problems & Imaging, 2010, 4 (4) : 639-647. doi: 10.3934/ipi.2010.4.639
##### References:
 [1] L. Ehrenpreis, "Fourier Analysis in Several Complex Variables,", Wiley & Sons, (1970).   Google Scholar [2] L. Ehrenpreis, "The Universality of the Radon Transform,", Oxford University Press, (2003).  doi: doi:10.1093/acprof:oso/9780198509783.001.0001.  Google Scholar [3] L. Ehrenpreis, Hypergeometric functions,, in, I (1988), 85.   Google Scholar [4] H. Farkas and I. Kra, "Riemann Surfaces,", Springer-Verlag, (1992).   Google Scholar [5] E. W. Hobson, "The Theory of Spherical and Ellipsoidal Harmonics,", Cambridge University Press, (1931).   Google Scholar [6] E. G. Kalnins, "Separation of Variables for Riemannian Spaces of Constant Curvature,", Longman, (1986).   Google Scholar [7] W. Miller, Jr., "Symmetry and Separation of Variables,", Addison-Wesley Publ. Co., (1977).   Google Scholar [8] N. Ja. Vilenkin and A. U. Klimyk, "Representations of Lie Groups and Special Functions,", Kluwer Acad. Publ., (1991).   Google Scholar

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##### References:
 [1] L. Ehrenpreis, "Fourier Analysis in Several Complex Variables,", Wiley & Sons, (1970).   Google Scholar [2] L. Ehrenpreis, "The Universality of the Radon Transform,", Oxford University Press, (2003).  doi: doi:10.1093/acprof:oso/9780198509783.001.0001.  Google Scholar [3] L. Ehrenpreis, Hypergeometric functions,, in, I (1988), 85.   Google Scholar [4] H. Farkas and I. Kra, "Riemann Surfaces,", Springer-Verlag, (1992).   Google Scholar [5] E. W. Hobson, "The Theory of Spherical and Ellipsoidal Harmonics,", Cambridge University Press, (1931).   Google Scholar [6] E. G. Kalnins, "Separation of Variables for Riemannian Spaces of Constant Curvature,", Longman, (1986).   Google Scholar [7] W. Miller, Jr., "Symmetry and Separation of Variables,", Addison-Wesley Publ. Co., (1977).   Google Scholar [8] N. Ja. Vilenkin and A. U. Klimyk, "Representations of Lie Groups and Special Functions,", Kluwer Acad. Publ., (1991).   Google Scholar
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