For a Fuchsian system $dY/dx=({\sum }_{j=}^p\left(A_j\right)/(x-t_j))Y$, (F) $t₁,t₂,...,t_p$ being distinct points in ℂ and $A₁,A₂,...,A_p\in M(n\times n;ℂ)$, the number α of accessory parameters is determined by the spectral types $s\left(A₀\right),s\left(A₁\right),...,s\left(A_p\right)$, where $A₀=-{\sum }_{j=1}^p{A}_j$. We call the set $z=(z₁,z₂,...,z_{\alpha })$ of α parameters a regular coordinate if all entries of the $A_j$ are rational functions in z. It is not yet known that, for any irreducibly realizable set of spectral types, a regular coordinate does exist. In this paper we study a process of obtaining a new regular coordinate from a given one by a coalescence of eigenvalues of the matrices...

2000 MSC: 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05.After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding...