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Hypergeometric functions over finite fields. (English) Zbl 0629.12017

This is an interesting contribution to recent efforts of finding character sum analogs over finite fields for various classical special functions. The starting point is the definition of the binomial coefficient \(\binom{A}{B}= (1/q) B(-1) J(A,\bar B)\) for multiplicative characters \(A\), \(B\) of the finite field \(\mathbb{F}_q\) of order \(q\), where \(J(A,\bar B)\) is the Jacobi sum for \(A\) and the conjugate character \(\bar B\) of \(B\). This leads to analogs of the binomial theorem and of standard identities for binomial coefficients.
Hypergeometric functions over \(\mathbb{F}_q\) are then defined in analogy with the power series expansions of classical generalized hypergeometric functions. Analogs of transformation formulas such as those of Pfaff and Euler and analogs of summation theorems such as Saalschütz’s theorem and Dixon’s theorem are established.
{It should be noted that the reference for Whipple’s theorem on p. 96 is incorrect.}

MSC:

11T24 Other character sums and Gauss sums
33C05 Classical hypergeometric functions, \({}_2F_1\)
11L05 Gauss and Kloosterman sums; generalizations
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