Certain transformations for hypergeometric series in the \(p\)-adic setting. (English) Zbl 1365.11121
Summary: In [Pac. J. Math. 261, No. 1, 219–236 (2013; Zbl 1296.11079)], D. McCarthy defined a function \(_nG_n[\cdots]\) using the Teichmüller character of finite fields and quotients of the \(p\)-adic gamma function. This function extends hypergeometric functions over finite fields to the \(p\)-adic setting. In this paper, we give certain transformation formulas for the function \(_nG_n[\cdots]\) which are not implied from the analogous hypergeometric functions over finite fields.
MSC:
11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
33E50 | Special functions in characteristic \(p\) (gamma functions, etc.) |
11T24 | Other character sums and Gauss sums |
11G20 | Curves over finite and local fields |
Keywords:
character of finite fields; Gaussian hypergeometric series; elliptic curves; trace of Frobenius; Teichmüller character; \(p\)-adic gamma functionCitations:
Zbl 1296.11079References:
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