In [3] André showed that the minimal differential equations of Э-functions and E-functions have a basis of holomorphic solutions at every point except for zero and infinity. In addition, in [5] he observed that if $f_1(z)$ is an E-function with rational coefficients such that $f_1(1)=0$, then $f_1(z)/(z-1)$ is also an E-function. With this additional result André derived transcendence results for the values of E-functions. These results were further applied by Beukers [7] to obtain a strengthened Siegel-Shidlovskii theorem for E-functions. The arguments of André and Beukers should aid in obtaining a Siegel-Shidlovskii type theorem for Э-functions provided that one can show that if $f_{-1}(z)$ is a Э-function with rational coefficients such that the 1-summation $F_{-1}(z)$ vanishes at $z=1$, then $f_{-1}(z)/(z-1)$ is also a Э-function. In this paper, we investigate this problem and derive several criteria to check its validity.

在 [3] 中，André 表明 Э 函数和E函数的最小微分方程在除零和无穷大之外的每一点都有全纯解的基。此外，在[5]中他观察到，如果$F_1(z)$是一个具有有理系数的E函数，使得$F_1(1)=0$， 然后 $F_1(z)/(z-1)$也是一个E函数。有了这个额外的结果，安德烈推导出了E函数值的超越结果。Beukers [7] 进一步应用了这些结果，以获得E函数的强化 Siegel-Shidlovskii 定理。André 和 Beukers 的论证应该有助于获得 Э 函数的 Siegel-Shidlovskii 型定理，前提是可以证明如果$F_{-1}(z)$ 是一个具有有理系数的 Э-函数，使得 1-和 $F_{-1}(z)$ 消失于 $z=1$， 然后 $F_{-1}(z)/(z-1)$也是一个 Э 函数。在本文中，我们调查了这个问题并得出了几个标准来检查其有效性。