Borweins’ cubic theta functions revisited

Autor: Heng Huat Chan, Liuquan Wang
Rok vydání: 2021
Předmět:
Zdroj: The Ramanujan Journal. 57:55-70
ISSN: 1572-9303
1382-4090
DOI: 10.1007/s11139-021-00503-x
Popis: Around 1991, J. M. Borwein and P. B. Borwein introduced three cubic theta functions a(q), b(q) and c(q) and discovered many interesting identities associated with these functions. The cubic theta functions b(q) and c(q) have product representations and these representations were first established using the theory of modular forms. The first elementary proof of the product representation of b(q) was discovered in 1994 by the Borweins and F. G. Garvan using one of Euler’s identity. They then derived the product representation of c(q) using transformation formulas of Dedekind’s $$\eta (\tau )$$ and some elementary identities satisfied by a(q), b(q) and c(q). In this note, we present three proofs of the product representation of c(q) without the use of the transformation of Dedekind’s $$\eta $$ -function. We also discuss the connections between these proofs and the works of Baruah and Nath (Proc Am Math Soc 142:441–448, 2014) and Ye (Int J Number Theory 12(7):1791–1800, 2016). We also adopt the idea of the Borweins and Garvan to derive the product representation of Jacobi theta function $$\vartheta _4(0|\tau )$$ which leads to a proof of the Jacobi triple product identity.
Databáze: OpenAIRE
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