q-Analogues of some supercongruences related to Euler numbers

Let E-n be the nth Euler number and (a)(n) = a(a + 1) ... (a + n - 1) the rising factorial. Let p > 3 be a prime. In 2012, Sun proved that Sigma((p-1)/2)(k=0)(-1)(k)(4k + 1)(1/2)(k)(3)/k!(3) equivalent to p(-1)((p-1)/2) + p(3)E(p-3) (mod p(4)), which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: Sigma(p-1)(k=0)(-1)(k)(3k + 1)1/2(k)(3)/k!(3) equivalent to p(-1)((p-1)/2) + p(3)E(p-3) (mod p(4)), which was originally conjectured by Sun. In this paper, we give q-analogues of the above two supercongruences as well as another supercongruence related to Euler numbers by employing the q-WZ method. As a conclusion, we also provide a q-analogue of the following supercongruence of Sun: Sigma((p-1)/2)(k=0)(1/2)(k)(2)/k!(2) equivalent to (-1)((p-1)/2) + p(2)E(p-3) (mod p(3)).

Automated summary

The text discusses several famous number theory equations and their applications and generalizations in mathematical research, including Euler numbers, supercongruences, etc.