Consider a differential equation $$ \delta^n+b_1(z)\delta^{n-1}+\ldots+b_n(z) $$ with a regular singular point zero. (Here $\delta=z\frac\partial{\partial z}$). Assume that its indicial polynomial $\mu^n+b_1(0)\mu^{n-1}+\ldots+b_n(0)$ has distinct integer roots $\mu_1<\mu_2<\ldots<\mu_n$ (thus the equation is resonant at zero).
If the monodromy is trivial, then near zero there is a basis of solutions of the form $z^{\lambda_i}f_i(z)$, where $f_i(z)$ is holomorphic near zero and $f_i(0)\ne0$. Is it true that $\lambda_i$ is necessarily a permutation of $\mu_i$?
What can one say if the monodromy is non-trivial?
