Borweins’ cubic theta functions revisited

ound 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 η (τ ) 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 η -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 ϑ _4(0|τ ) which leads to a proof of the Jacobi triple product identity.