## 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, $${}_pF_q$$