Modular Proofs of Gosper's Identities

Abstract

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 $m\ge 3$, ${\Pi }_{q^{n_1}}$, $\cdots$, ${\Pi }_{q^{n_m}}$ 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.

22 pages. Minor changes may be made after v2