Ramanujan-type series for 1/π involving S_n^(2)

Prove the two Ramanujan-type series identities involving S_k^{(2)}(1/11) and S_k^{(2)}(1/18) displayed in equations (1.13) and (1.14), respectively.

Background

The paper seeks new series for 1/π built from the polynomial family S_n^{(m)}(x). The two identities in Conjecture 1.2 were checked numerically and are related to later p-adic congruence conjectures.