After some remarks about the analogy between the classical gamma-function and Gaussian sums over finite fields a complete, very short explicit proof is given of an identity expressing a certain sum of products of Gaussian sums as a product of Gaussian sums. This identity is an analogue of the classical Barnes’ first lemma for the gamma-function. Four multiplicative characters of a finite field are concerned; the usually necessary restrictions on the triviality of certain products of these characters are avoided by the use of corrective terms. References are given for other approaches of this identity.In [2] a parallel proof is given for the classical identity and its finite analogue; the status of this reference has meanwhile changed from “preprint” to “published”: Can. Math. Bull. 36, No. 3, 273-282 (1993; Zbl 0803.33001), the status of reference [4] has changed from “to appear” into “published”: Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 179-188 !
@article{701518,
title = {Gamma-function and Gaussian-sum-function},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
series = {GDML\_Books},
publisher = {Circolo Matematico di Palermo},
address = {Palermo},
year = {1993},
pages = {[195]-200},
mrnumber = {MR1246633},
zbl = {0806.11060},
url = {http://dml.mathdoc.fr/item/701518}
}
Helversen-Pasotto, A. Gamma-function and Gaussian-sum-function, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1993), pp. [195]-200. http://gdmltest.u-ga.fr/item/701518/