Modular proofs of Gosper's identities

We give unified modular proofs to all of Gosper's identities on the q-constant Πq. also confirm observation that for any distinct positive integers n1,⋯,nm with m≥3, Πqn1, ⋯, Πqnm satisfy a nonzero homogeneous polynomial. Our provide method rediscover identities. Meanwhile, several results Πq found by El Bachraoui have been revised. Furthermore, we illustrate strategy construct some using hauptmoduls genus zero congruence subgroups.