Modular Proofs of Gosper's Identities

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