Congruences involving S_k^(2)(1/7)

Establish the mod-p^2 congruences and higher-power p-adic relations involving S_k^{(2)}(1/7) stated in Conjecture 4.5.

Background

Conjecture 4.5 gives supercongruences for S_k{(2)}(1/7), including a weighted sum closely related to the conjectural Ramanujan-type series in Conjecture 1.2.

References

Conjecture 4.5. Let p > 7 be a prime.

— A family of polynomials and related congruences and series  (2505.02767 - Sun, 5 May 2025) in Conjecture 4.5, Section 4