A new extension of the (H.2) supercongruence of Van Hamme for primes \(p\equiv 3\operatorname{mod}{4}\) (Q2117549)

Using Andrews' multiseries generalization of Watson's \({}_8\phi_7\) transformation, the author gives a new extension of the (H.2) supercongruence of Van Hamme. Theorem 1. Let \(s\ge 0\) be an even integer. Let \(p\) be an odd prime with \(p\ge 2s+3\) and \(p\equiv 3\pmod 4\). Then \[\sum_{k=0}^{(p-1)/2}(4k+1)^s\frac{(\frac12)_k^3}{k!^3}\equiv0\pmod{p^2}.\] The author proves Theorem 1 by establishing its \(q\)-analogue. Theorem 2. Let \(s\ge 0\) be an even integer. Let \(n\) be an odd integer with \(n\ge 2s+3\) and \(n\equiv 3\pmod 4\). Then \[\sum_{k=0}^{(n-1)/2}[4k+1]_{q^2}[4k+1]^{s-1}\frac{(q^2;q^4)_k^3}{(q^4;q^4)_k^3}q^{(1-2s)k}\equiv 0\pmod{\phi_n(q)^2}.\] The author also gives a generalization of a congruence which was proved by Mao and Pan, as well as its \(q\)-analogue.