# 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
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##### References:
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