Modular Proofs of Gospers Identities

We give unified modular proofs to all of Gospers identities on the $q$-constant $Pi_q$. We also confirm Gospers observation that for any distinct positive integers $n_1,cdots,n_m$ with $mgeq 3$, $Pi_{q^{n_1}}$, $cdots$, $Pi_{q^{n_m}}$ satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gospers identities. Meanwhile, several results on $Pi_q$ found by El Bachraoui have been corrected. Furthermore, we illustrate a strategy to construct some of Gospers identities using hauptmoduls for genus zero congruence subgroups.