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 and Imaging, 2010, 4 (4) : 639-647. doi: 10.3934/ipi.2010.4.639
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L. Ehrenpreis, "Fourier Analysis in Several Complex Variables," Wiley & Sons, Interscience, 1970.

[2]

L. Ehrenpreis, "The Universality of the Radon Transform," Oxford University Press, 2003. doi: doi:10.1093/acprof:oso/9780198509783.001.0001.

[3]

L. Ehrenpreis, Hypergeometric functions, in "Algebraic Analysis," vol. I, Academic Press, New York, (1988), 85-128.

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H. Farkas and I. Kra, "Riemann Surfaces," Springer-Verlag, 1992.

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E. W. Hobson, "The Theory of Spherical and Ellipsoidal Harmonics," Cambridge University Press, 1931.

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E. G. Kalnins, "Separation of Variables for Riemannian Spaces of Constant Curvature," Longman, Sci. Tech., Wiley & Sons, New York 1986.

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W. Miller, Jr., "Symmetry and Separation of Variables," Addison-Wesley Publ. Co., Reading, Mass., 1977.

[8]

N. Ja. Vilenkin and A. U. Klimyk, "Representations of Lie Groups and Special Functions," Kluwer Acad. Publ., Dortrecht, Netherlands, 1991.

show all references

References:
[1]

L. Ehrenpreis, "Fourier Analysis in Several Complex Variables," Wiley & Sons, Interscience, 1970.

[2]

L. Ehrenpreis, "The Universality of the Radon Transform," Oxford University Press, 2003. doi: doi:10.1093/acprof:oso/9780198509783.001.0001.

[3]

L. Ehrenpreis, Hypergeometric functions, in "Algebraic Analysis," vol. I, Academic Press, New York, (1988), 85-128.

[4]

H. Farkas and I. Kra, "Riemann Surfaces," Springer-Verlag, 1992.

[5]

E. W. Hobson, "The Theory of Spherical and Ellipsoidal Harmonics," Cambridge University Press, 1931.

[6]

E. G. Kalnins, "Separation of Variables for Riemannian Spaces of Constant Curvature," Longman, Sci. Tech., Wiley & Sons, New York 1986.

[7]

W. Miller, Jr., "Symmetry and Separation of Variables," Addison-Wesley Publ. Co., Reading, Mass., 1977.

[8]

N. Ja. Vilenkin and A. U. Klimyk, "Representations of Lie Groups and Special Functions," Kluwer Acad. Publ., Dortrecht, Netherlands, 1991.

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