Motivated by arithmetic properties of partition numbers $p\left(n\right)$, our goal is to find algorithmically a Ramanujan type identity of the form $\sum _{n=0}^{\mathrm{\infty }}p(11n+6)q^n=R$, where $R$ is a polynomial in products of the form $e_{\alpha }:=\prod _{n=1}^{\mathrm{\infty }}(1-q^{{11}^{\alpha }n})$ with $\alpha =0,1,2$. To this end we multiply the left side by an appropriate factor such the result is a modular function for ${\mathrm{\Gamma }}_0\left(121\right)$ having only poles at infinity. It turns out that polynomials in the $e_{\alpha }$ do not generate the full space of such functions, so we were led to modify our goal. More concretely, we give three different ways to construct the space of modular functions for ${\mathrm{\Gamma }}_0\left(121\right)$ having only poles at infinity. This in turn leads to three different representations of $R$ not solely in terms of the $e_{\alpha }$ but, for example, by using as generators also other functions like the modular invariant $j$.

由分区数的算术特性驱动 $磷\left(n\right)$，我们的目标是在算法上找到形式的拉马努金类型身份 $\sum _{n=0}^{\mathrm{\infty }}磷(11n+6)q^n=\mathrm{电阻}$， 在哪里 $\mathrm{电阻}$ 是形式为乘积的多项式 ${\mathrm{电子}}_{\alpha }:=\prod _{n=1}^{\mathrm{\infty }}(1-q^{{11}^{\alpha }n})$ 和 $\alpha =0,1,2$. 为此，我们将左侧乘以一个适当的因子，结果是一个模函数${\mathrm{\Gamma }}_0\left(121\right)$在无穷远处只有极点。事实证明，多项式在${\mathrm{电子}}_{\alpha }$没有生成这些函数的完整空间，所以我们被引导修改了我们的目标。更具体地说，我们给出了三种不同的方式来构造模函数空间${\mathrm{\Gamma }}_0\left(121\right)$在无穷远处只有极点。这反过来又导致了三种不同的表示$\mathrm{电阻}$ 不仅仅是在 ${\mathrm{电子}}_{\alpha }$ 但是，例如，通过将其他函数用作生成器，例如模不变量 $j$.