Modular Proofs of Gosper's Identities

DOI10.1016/J.AAM.2021.102312arXiv2108.06319WikidataQ113881577 ScholiaQ113881577MaRDI QIDQ6375186

Liuquan Wang

Publication date: 13 August 2021

Abstract: We give unified modular proofs to all of Gosper's identities on the $q$-constant $P{i}_q$. We also confirm Gosper's observation that for any distinct positive integers $n_1,cdots,n_m$ with $mgeq3$, $P{i}_{q^{n_1}}$, $cdots$, $P{i}_{q^{n_m}}$ satisfy a nonzero homogeneous polynomial. Our proofs provide a method to rediscover Gosper's identities. Meanwhile, several results on $P{i}_q$ 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.

Mathematics Subject Classification ID

Modular and automorphic functions (11F03) Basic hypergeometric functions in one variable, ({}_rphi_s) (33D15) Holomorphic modular forms of integral weight (11F11)

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