Proofs for certain q-trigonometric identities of Gosper
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Applying an addition formula of Liu (2007), we deduce certain Jacobi theta function identities. From these results we confirm several q-trigonometric identities conjectured by Gosper (2001). Another conjectured identity on the constant Πq is also settled.
KeywordsJacobi theta function addition formula q-trigonometric identity constant Πq
MSC(2010)11F27 33E05 11E25
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The first author was supported by National Natural Science Foundation of China (Grant No. 11801451). The second author was supported by National Natural Science Foundation of China (Grant No. 11371184) and the Natural Science Foundation of Henan Province (Grant Nos. 162300410086, 2016B259 and 172102410069). The authors thank the referees for their careful reading of the manuscript and helpful comments.