Abstract
A new proof of the triplication formula for Gauss sums is given. It mimics an old proof of the analogous result for gamma functions. The techniques are formal and rely upon the character properties of fields. A new character sum evaluation is given.
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References
Askey, R.,The q-gamma and q-beta functions. Applicable Anal.8 (1978/79), 125–141.
Boyarsky, M.,p-adic gamma functions and Dwork cohomology. Trans. Amer. Math. Soc.257 (1980), 359–568.
Davenport, H. andHasse, H.,Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklishen Fällen. J. Reine Angew. Math.172 (1934), 151–182.
Edwards, J.,Integral calculus, vol. 2. Macmillan, London, 1922.
Erdélyi, A.,Higher transcendental functions, vol. I. McGraw Hill, New York-Toronto-London, 1953.
Gel'fand, I., Graev, M., andPyateckii-sapiro, I.,Representation theory and automorphic functions. Saunders, 1969.
Gross, B. andKoblitz, N.,Gauss sums and the p-adic γ-function. Ann. of Math. (2)109 (1979), 569–581.
Ireland, K. andRosen, M.,Elements of number theory. Bogden and Quigley, Tarrytown-on-Hudson, 1972.
Jacobi, C.,Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie. J. Reine Angew. Math.30 (1846), 166–182.
Koblitz, N.,q-extensions of the p-adic gamma function. Trans. Amer. Math. Soc.260 (1980), 449–457.
Koblitz, N.,q-extensions of the q-adic gamma function, II. Trans. Amer. Math. Soc.273 (1982), 111–130.
Liouville, J.,Détermination des valeurs d'une class remarquable d'intégrales definies multiples, et demonstration nouvelle d'une cèlébre formule de Gauss concernant les fonctions gamma de Legendre. J. Math. (2nd series)1 (1856), 82–88.
Sears, D.,On the transformation theory of basic hypergeometric functions. Proc. London Math. Soc. (2)53 (1951), 158–180.
Yamamoto, K.,On a conjecture of Hasse concerning multiplicative relations of Gaussian sums. J. Combin. Theory Ser.A 1 (1966), 476–489.
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Greene, J., Stanton, D. The triplication formula for Gauss sums. Aeq. Math. 30, 134–141 (1986). https://doi.org/10.1007/BF02189920
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DOI: https://doi.org/10.1007/BF02189920