It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in
the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:

 Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.

If $d\neq 0$, the system is not Fuchsian: singularity at $\infty$ is irregular. If $k=2, d\neq 0$ your system already contains the 
"prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy
is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is

Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.