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Some conjectures concerning partial sums of generalized hypergeometric series. (English) Zbl 0895.11051

Schikhof, W. H. (ed.) et al., \(p\)-adic functional analysis. Proceedings of the fourth international conference, Nijmegen, Netherlands, June 3–7, 1996. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 192, 223-236 (1997).
This paper is the result of a computer search for \(p\)-adic analogs of hypergeometric series identities. The author discovered thirteen congruences involving the \(p\)-adic gamma function of which the following is typical \((p\neq 2)\):
\[ \sum_{k=0}^{(p-1)/2} (4k+1) \binom{-1/2}{k}^3 \equiv {-p \over \Gamma_p(1/2)^2} \pmod{p^3}. \]
The author has proven three of the thirteen identities, verified all congruences for all primes \(< 300\), and proven weaker results. Connections with modular forms are discussed.
For the entire collection see [Zbl 0869.00037].

MSC:

11A07 Congruences; primitive roots; residue systems
11-04 Software, source code, etc. for problems pertaining to number theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
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