Congruences involving S_k^(2)(1/2)

Establish all the mod-p^2 congruences and associated p-adic integrality assertions for sums involving S_k^{(2)}(1/2) stated in Conjecture 4.1, including the cases distinguished by representations p=x^2+y^2 and by p modulo 4 and 8.

Background

Section 4 collects conjectural supercongruences for truncated sums formed from the newly introduced polynomial family. Conjecture 4.1 concerns S_k^{(2)}(1/2), with separate formulas according to quadratic representations of the prime.