Modular proofs of Gosper's identities

Abstract

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