Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hypergeometric functions over finite fields


Author: John Greene
Journal: Trans. Amer. Math. Soc. 301 (1987), 77-101
MSC: Primary 11T21; Secondary 11L05, 33A35
MathSciNet review: 879564
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Abstract: In this paper the analogy between the character sum expansion of a complex-valued function over $ {\text{GF}}(p)$ and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.


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  • [1] Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145
  • [2] W. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge, 1935.
  • [3] H. Davenpōrt and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1934), 151-182.
  • [4] A. Erdélyi, Higher transcendental functions, Vol. 1, McGraw-Hill, New York, 1935.
  • [5] Ronald J. Evans, Identities for products of Gauss sums over finite fields, Enseign. Math. (2) 27 (1981), no. 3-4, 197–209 (1982). MR 659148
  • [6] Ronald J. Evans, Character sum analogues of constant term identities for root systems, Israel J. Math. 46 (1983), no. 3, 189–196. MR 733348, 10.1007/BF02761951
  • [7] Ronald J. Evans, Hermite character sums, Pacific J. Math. 122 (1986), no. 2, 357–390. MR 831119
  • [8] R. J. Evans, J. R. Pulham, and J. Sheehan, On the number of complete subgraphs contained in certain graphs, J. Combin. Theory Ser. B 30 (1981), no. 3, 364–371. MR 624553, 10.1016/0095-8956(81)90054-X
  • [9] J. Greene, Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields, Ph.D. thesis, Univ. of Minnesota, Minneapolis, 1984.
  • [10] J. Greene and D. Stanton, A character sum evaluation and Gaussian hypergeometric series, J. Number Theory 23 (1986), no. 1, 136–148. MR 840021, 10.1016/0022-314X(86)90009-0
  • [11] Anna Helversen-Pasotto, L’identité de Barnes pour les corps finis, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 6, A297–A300 (French, with English summary). MR 0476707
  • [12] Kenneth F. Ireland and Michael I. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR 661047
  • [13] C. Jacobi, Über die reisteilung und ihre Anwendung auf die Zahlentheorie, J. Reine Angew. Math. 30 (1846), 166-182.
  • [14] Neal Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math. 48 (1983), no. 1, 3–23. MR 700577
  • [15] Wen-Ch’ing Winnie Li and Jorge Soto-Andrade, Barnes’ identities and representations of 𝐺𝐿(2). I. Finite field case, J. Reine Angew. Math. 344 (1983), 171–179. MR 716253
  • [16] Rudolf Lidl and Harald Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
  • [17] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
  • [18] E. Whittaker and G. Watson, Modern analysis, Cambridge Univ. Press, Cambridge, 1947.

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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0879564-8
Article copyright: © Copyright 1987 American Mathematical Society