论文标题
Ramanujan的theta-funnctions a_ {m，n}的非凡产物的明确评估
The explicit evaluations of Ramanujan's remarkable product of theta-functions a_{m, n}
论文作者
论文摘要
在他的第一本笔记本中，Ramanujan在第338和339页上定义了theta-funnctions $ a_ {m，n} $的非凡产品。此外，他还记录了十八个显式值，具体取决于两个参数，即$ m $和$ n $，其中这些是奇怪的整数。所有这些值都是由Berndt等人确定的。在本文中，我们开始研究$ a_ {m，n} $的明确评估，即$ m $的值。我们为$ a_ {m，n} $的明确评估建立了一个新的通用定理，该评估涉及Ramanujan的类别不变。为此，我们建立了几个新的P-Q混合模块化方程。使用这些定理，我们计算了$ n = 3、5、7、13 $甚至$ m $的几个新的显式值$ a_ {m，n} $。
On pages 338 and 339 in his first notebook, Ramanujan defined the remarkable product of theta-functions $a_{m, n}$. Also he recorded eighteen explicit values depending on two parameters, namely, $m$, and $n$, where these are odd integers. All these values were established by Berndt et al. In this paper, we initiate to study explicit evaluations of $a_{m, n}$ for even values of $m$. We establish a new general theorems for the explicit evaluations of $a_{m, n}$ involving class invariant of Ramanujan. For this purpose, we establish several new P-Q mixed modular equations. Using these theorems, we calculate several new explicit values of $a_{m, n}$ for $n = 3, 5, 7, 13$ and even values of $m$.
