Bases in the solution space of the Mellin system

A. Dickenstein ; T. M. Sadykov

Sbornik. Mathematics, Tome 198 (2007) no. 9, pp. 1277-1298

Résumé

We consider algebraic functions $z$ satisfying equations of the following form: \begin{equation*} a_0 z^m+a_1z^{m_1}+a_2 z^{m_2}+\dots+a_nz^{m_n}+a_{n+1}=0. \tag{1} \end{equation*} Here $m>m_1>\dots>m_n>0$, $m,m_i\in\mathbb N$, and $z=z(a_0,\dots,a_{n+1})$ is a function of the complex variables $a_0,\dots,a_{n+1}$. Solutions of such algebraic equations are known to satisfy holonomic systems of linear differential equations with polynomial coefficients. In this paper we investigate one such system, which was introduced by Mellin. The holonomic rank of this system of equations and the dimension of the linear space of its algebraic solutions are computed. An explicit base in the solution space of the Mellin system is constructed in terms of roots of (1) and their logarithms. The monodromy of the Mellin system is shown to be always reducible and several results on the factorization of the Mellin operator in the one-variable case are presented. Bibliography: 18 titles.