Product identities in two variables x, q expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi’s triple product identity, Watson’s quintuple identity, and Hirschhorn’s septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers–Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers–Ramanujan identities; this amounts to an unexpected proportionality of canonical basis vectors, two of which can be viewed as two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers–Ramanujan identities, related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized eta functions and mock theta functions.

两个变量x，  q中的产品恒等式将无限乘积扩展为无限和，它们是θ函数的线性组合；著名的例子包括Jacobi的三重产品身份，Watson的五重身份和Hirschhorn的七重身份。我们将这些级数展开视为拟周期亚纯函数的某些向量空间（与线和向量束的截面有关）的规范基础中的表示，并找到两个非乘积，一个非乘积和几个两个变量罗杰斯–拉曼努扬的新恒等式输入总和。我们的主要定理解释了七元乘积标识与两个原始的Rogers-Ramanujan标识之间的对应关系。这等于规范基础向量的意外比例，其中两个可以看作是五阶模拟theta函数的二变量类似物。我们还证明了Ewell的八乘积恒等式与Rogers-Ramanujan恒等式的两个简单变体之间的相似对应关系，它们与三阶模拟theta函数有关，并推测了这种现象的其他发生。作为应用程序，我们专用于我们的结果以获取广义eta函数和模拟theta函数的商的恒等式。