Hypergeometric Functions

  • Leon Ehrenpreis


Hypergeometric functions pervade many branches of mathematics. Perhaps the reason for this is that hypergeometric functions represent a confluence of three fundamental viewpoints
  1. a)

    Partial differential Operators and Lie groups

  2. b)

    Power series and ordinary differential operators.

  3. c)

    Algebraic integrals.

In this paper we shall be concerned mainly with (a) and (b). Section II centers around the ideas of (a) and how to go from (a) to (b). In Section III we show how to reverse this process and derive (a) from (b). This involves an analysis of new methods of factorization which we call Hadamard-Hermite factorization. We explain how the q analog of Hadamard-Hermite factorization leads to a proof of the Rogers -Ramanujan identities of partition theory.


Hypergeometric Function Weyl Group Parabolic Subgroup Partial Differential Operator Conformal Group 


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  1. 1.
    L. Ehrenpreis, Fourier Analysis in Several Comylex Variables. Wiley-Interscience, New York (1970).Google Scholar
  2. 2.
    “Hyper geometric Functions”, in, Algebraic Analysis Vol. I (1988), Academic Press, New York.Google Scholar
  3. 3.
    “Function theory for Rogers-Ramanjan-like partition identities”, to appear in Springer volume dedicated to the memory of Emil Grosswald..Google Scholar

Copyright information

© Springer-Verlag Tokyo 1991

Authors and Affiliations

  • Leon Ehrenpreis
    • 1
  1. 1.BrooklyUSA

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