Cubic Analogues of the Jacobian Theta Function θ(z, q)

Authors:
  • MHMichael Hirschhorn
    Michael Hirschhorn
  • FGFrank Garvan
    Frank Garvan
  • JBJon Borwein
    Jon Borwein
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Abstract
There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity