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

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

Congruences modulo 9 and 27 for overpartitions

“…Proof Hirschhorn et al [10] proved that By (3.6) and mathematical induction, we find that for n ≥ 0 and k ≥ 0,…”

Cubic Identities for Theta Series in Three Variables

“…(using (4)) which is (1.25) in [3]. A variant of the argument of Theorem 1 gives the following result.…”

“…We call these functions theta series for convenience. Subsequently Hirschhorn, Garvan and J. Borwein [3] proved the corresponding identity for two-variable analogues of these theta series. Solé [4] (see also [5]) gave a new proof of (1) using a lattice having the structure of a Z[ω]-module.…”

“…We show that these functions specialize to the two-variable functions introduced in [3]. First of all, each element α ∈ O can be uniquely written as α = nω − mω 2 .…”

Cubic elliptic functions

“…Similarly, we will abbreviate b(q, 1) and c(q, 1) to b(q) and c(q), respectively. The functions a (q, z), a(q, z), b(q, z) and c(q, z) were introduced 1 by Hirschhorn et al [13]. They showed [13, (1.22…”