McCarthy, Dermot; Osburn, Robert A \(p\)-adic analogue of a formula of Ramanujan. (English) Zbl 1175.33004 Arch. Math. 91, No. 6, 492-504 (2008). From the text: During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, L. van Hamme [Lect. Notes Pure Appl. Math. 192, 223–236 (1997; Zbl 0895.11051)] recently conjectured \(p\)-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.Conjecture 1.1. Let \(p\) be an odd prime. Then \[ \sum_{k=0}^{\frac{p-1}{2}} (4k + 1) \binom{-\frac12}{k}^5 \equiv \begin{cases} -\frac{p}{\Gamma_p(\frac34)^4} \pmod{p^3}\quad &\text{ if }p\equiv 1 \pmod 4 \\ \quad 0 \pmod{p^3} \quad &\text{ if }p\equiv 3 \pmod 4 \end{cases} \] where \(\Gamma_p(\cdot)\) is the \(p\)-adic Gamma function. Cited in 1 ReviewCited in 50 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11B65 Binomial coefficients; factorials; \(q\)-identities 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) 33C20 Generalized hypergeometric series, \({}_pF_q\) Keywords:Gaussian hypergeometric series; supercongruences Citations:Zbl 0895.11051 PDFBibTeX XMLCite \textit{D. McCarthy} and \textit{R. Osburn}, Arch. Math. 91, No. 6, 492--504 (2008; Zbl 1175.33004) Full Text: DOI arXiv