## 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, $${}_pF_q$$

### Keywords:

Gaussian hypergeometric series; supercongruences

Zbl 0895.11051
Full Text: