Séminaire Lotharingien de Combinatoire, B33f (1994), 15 pp.
Gaußsche Summen über endlichen Körpern und
Gaussian sums over finite fields are analogous to values of the
gamma-function as can be seen via the Eulerian integral for the
gamma-function and this seems to have been known to Gauss already;
indeed, this seems to have been for him one of the reasons to
introduce these sums. Nevertheless it has been astonishing for several
authors to discover identities for Gaussian sums which bear a strong
formal analogy with classical identities for the gamma function;
also a p-adic gamma function has been invented and many of the
well-known classical identities have been shown to admit p-adic
analogs. The aim of these lectures is to give an introduction
to the subject discussing some identities, giving some proofs
and pointing out relations to the representation theory
of the general linear group of a finite field (Hecke algebras).
The analogy between binomial coefficients and Jacobi sums leading to
hypergeometric functions over finite fields is mentioned.
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