advances in mathematics 128, 186189 (1997) Character Sum Identities in Analogy with Special Functions Identities Anna P. Helversen-Pasotto Laboratoire ``Jean-Alexandre Dieudonne ,'' U.R.A au C.N.R.S. 168, Departement de Mathematiques, U.F.R. Faculte des Sciences, Universite de Nice Sophia Antipolis, Parc Valrose, Bo@^te Postale 71, F-06108 Nice Cedex 02, France Received May 20, 1996; accepted December 30, 1996 In the beginning of the century Barnes found the identity 1 2?i | +i &i 1( a +s ) 1( b &s ) 1( c +s ) 1( d &s ) ds = 1( a +b) 1( b +c ) 1( c +d ) 1( a +d ) 1( a +b +c +d ) For a, b, c, d complex numbers such that none of a +b, b +c, c +d, a +d is 0 or a negative integer (pole of the gamma-function); the path of integra- tion is chosen such that the poles of 1( a +s ) 1( c +s ) are separated from those of 1( b &s )1( d &s ). For a proof see [Ba, HS, S, or WW]. Replacing the gamma-function 1 by the Gaussian-sum-function G we obtain the ``finite Barnes identity'' 1 q &1 : S G( AS) G( BS &1 ) G( CS) G( DS &1 ) = G( AB) G( BC) G( CD) G( AD) G( ABCD) , where A, B, C, D are elements of the group X of multiplicative characters of the finite field F q of q elements such that the product ABCD is not the trivial character; the summation is extended over all S in X; the Gaussian- sum-function is defined on X and the values are complex numbers; for T in X we have G( T )=: T( a) ( a) article no. AI971637 186 0001-870897 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.