Confluence of Fuchsian second-order differential equations

Ordinary linear homogeneous second-order differential equations with polynomial coefficients including one in front of the second derivative are studied. Fundamental definitions for these equations: of $s$-rank of the singularity (different from Poincaré rank), of $s$-multisymbol of the equation and of $s$-homotopic transformations are proposed. Generalization of Fuchs\rq theorem for confluent Fuchsian equations is proved. The tree structure of types of equations is exposed and the generalized confluence theorem is proved.