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A \(p\)-adic analogue of a formula of Ramanujan. (English) Zbl 1175.33004

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.

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, \({}_pF_q\)

Citations:

Zbl 0895.11051
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