Suppose $G_n(w)$ is a formal power series, try to solve out $G_n(w)$ for all $n\ge0$ from the equation: $$\sum_{n\ge0}z^nG_n(w)=we^z\sum_{n\ge0}\frac{!n}{n!}z^nG_n(w)+1-w\tag1$$ where $!n$ is $n$-subfactorial which satisifies $$\frac{!n}{n!}=\sum_{k=0}^n\frac{(-1)^k}{k!}$$ Any help? Thanks! Th...

Let the transformation rule between two coordinate systems $ (x_1, x_2, x_3) $, and $ (u_1, u_2, u_3) $ be $$ x_1 = a_{11} u_1 + a_{12} u_2 + a_{13} u_3 \\ x_2 = a_{21} u_1 + a_{22} u_2 + a_{23} u_3 \\ x_3 = a_{31} u_1 + a_{32} u_2 + a_{33} u_3 \\ $$ If $ \hat e_{x1}, \hat e_{x2}, \hat e_{x3}$ ...

Suppose $G_n(w)$ is a formal power series, try to solve out $G_n(w)$ for all $n\ge0$ from the equation: $$\sum_{n\ge0}z^nG_n(w)=we^z\sum_{n\ge0}\frac{!n}{n!}z^nG_n(w)+1-w\tag1$$ where $!n$ is $n$-subfactorial which satisifies $$\frac{!n}{n!}=\sum_{k=0}^n\frac{(-1)^k}{k!}$$ Any help? Thanks! Th...

Let the transformation rule between two coordinate systems $ (x_1, x_2, x_3) $, and $ (u_1, u_2, u_3) $ be $$ x_1 = a_{11} u_1 + a_{12} u_2 + a_{13} u_3 \\ x_2 = a_{21} u_1 + a_{22} u_2 + a_{23} u_3 \\ x_3 = a_{31} u_1 + a_{32} u_2 + a_{33} u_3 \\ $$ If $ \hat e_{x1}, \hat e_{x2}, \hat e_{x3}$ ...