论文标题
与两变量的Rogers-Ramanujan类型和模拟theta功能有关的几种新产品标识
Several new product identities in relation to two-variable Rogers-Ramanujan type sums and mock theta functions
论文作者
论文摘要
两个变量中的产品身份$ x，q $扩展了无限总和，这是theta函数的线性组合；著名的例子包括雅各比的三级产品身份，沃森的五重奏身份以及赫希霍恩的隔隔式身份。我们将这些串联扩展视为拟层近形函数的某些向量空间的规范底座中的表示形式（与线路和矢量捆绑包有关），并为两种非重心产品，一种不启用的产品找到新的身份，以及几种两种可变性的Rogers-Ramanujan类型Sums。我们的主要定理解释了分子产品身份与两个原始的Rogers-Ramanujan身份之间的对应关系，涉及五阶模拟theta函数的两变量类似物。我们还证明了Ewell的八枚产品身份与Rogers-Ramanujan身份的两个更简单的变化之间的相似对应关系，这与三阶模拟theta函数有关，并猜想了这种现象的其他情况。作为应用程序，我们专门研究结果，以获取广义Dedekind ETA功能和模拟theta功能的商的身份。
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, involving 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, which is 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 Dedekind eta functions and mock theta functions.
