Stanley引理与多重θ函数
Stanley's Lemma and Multiple Theta Functions
摘要：
本文提出了一种以θ函数乘积$[(-1)^δa_1^{α_1}a_2^{α_2}\cdots a_r^{α_r}q^{s}; q^{t}]_\infty$形式的多个θ函数恒等式的算法验证方法，其中$α_i$是整数，$δ=0$或$1$，$s\in \mathbb{Q}$，$t\in \mathbb{Q}^{+}$，且指数向量$(α_1,α_2,\ldots,α_r)$在$\mathbb{Q}$上线性无关.对于这样的多个θ函数的身份，我们提供了一个算法的方法来计算所有涉及的多个θ函数所满足的连续关系的系统。利用斯坦利引理的基本平行六面体，我们表明，一个多θ函数可以由有限数量的系数。因此，这样的恒等式可以简化为有限个更简单的关系。许多经典的多theta函数恒等式都属于这个框架，包括黎曼加法公式和推广的黎曼恒等式。
英文摘要：
We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions $[(-1)^δa_1^{α_1}a_2^{α_2}\cdots a_r^{α_r}q^{s}; q^{t}]_\infty$, where $α_i$ are integers, $δ=0$ or $1$, $s\in \mathbb{Q}$, $t\in \mathbb{Q}^{+}$, and the exponent vectors $(α_1,α_2,\ldots,α_r)$ are linearly independent over $\mathbb{Q}$. For an identity on such multiple theta functions, we provide an algorithmic approach for computing a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's Lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a finite number of its coefficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity.
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