Abstract
In 2001 W. Gosper introduced constant Πq, and conjectured many intriguing identities on this constant. In this paper we employ some knowledge of modular equations with degree 5 to confirm several of Gosper's Πq-identities. Two strange q-identities involving Πq and a Lambert series conjectured by W. Gosper are proved. As a consequence, a q-identity involving Πq and two Lambert series, which was conjectured by Gosper, is proved. As an application, we confirm an interesting q-trigonometric identity of Gosper, which can be viewed as a theta function analogue for the related well-known trigonometric identity. Our proof relies on an interesting general theta function identity of Zhi-Guo Liu.
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