论文标题
第二个新的P-Q模块化二级的应用
Applications of a new P-Q modular equation of degree two
论文作者
论文摘要
在他的第一本笔记本中，Ramanujan在分散的地方记录了他们满足的107个级别不变的值或不可约的一元多项式。在他的第二本笔记本中，在第294-299页的第294-299页上，他为77个班级不变的$ g_n $和$ g_n $的价值表提供了一张。传统上，$ g_n $是针对$ n $的奇数和$ g_n $确定的，即$ n $。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} $的非凡产品。此外，他根据两个参数（即$ m $和$ n $）记录了18个显式值，其中这些是奇怪的整数。在本文中，我们开始研究$ g_n $的明确评估，即$ n $。我们为$ g_n $的明确评估建立了一个新的通用公式，该评估涉及类不变$ g_n $。为此，我们得出了第二级的新P-Q模块化方程。该模块化方程的进一步应用，我们建立了一个新公式，以明确评估$ a_ {m，2} $。另外，我们计算了类别不变$ g_ {n} $的几个明确值和奇异模量$α_n$。
At scattered places in his first notebook, Ramanujan recorded the values for 107 class invariants or irreducible monic polynomials satisfied by them. On pages 294-299 in his second notebook, he gave a table of values for 77 class invariants $G_n$ and $g_n$ in his second notebook. Traditionally, $G_n$ is determined for odd values of $n$ and $g_n$ for even values of $n$. 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. In this paper, we initiate to study explicit evaluations of $G_n$ for even values of $n$. We establish a new general formula for the explicit evaluations of $G_n$ involving class invariant $g_n$. For this purpose, we derive a new P-Q modular equation of degree two. Further application of this modular equation, we establish a new formula to explicit evaluation of $a_{m, 2}$. Also, we compute several explicit values of class invariant $g_{n}$ and singular moduli $α_n$.
