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On Lamé operators which are pull-backs of hypergeometric ones. (English) Zbl 0851.34024
The author considers the Lamé operators (i.e. the differential operators of second-order with four singular points) having the form \[ L_n= D^2+ {f'\over 2f} D- {n(n+ 1) x+ B\over f},\tag{\(*\)} \] and tries to give a method which allows to calculate the number of these operators having prescribed monodromy. In \((*)\) \(D\) stands for \(d/dx\) while \(n\in \mathbb{N}\) and \(B\in \mathbb{C}\) are constants. The function \(f\) is given by \(f= 4(x- e_1) (x- e_2) (x- e_3)\), where \(e_i\in \mathbb{C}\) are constant numbers which differ from each other. He shows that the number (up to homography) of equations whose monodromy group is a dihedral group of order \(2m\) \((m\in \mathbb{N})\) is finite for each \(n\in \mathbb{N}\) and \(m\in \mathbb{N}\). Notice that two Lamé operators are homographic if one of them can be transformed into the other by a homographic change of the independent variable. The number of the operators (up to homography) is calculated for the case where \(n= 1\). The case of operators with infinite monodromy is also considered. The results are given in one lemma, nine propositions, six theorems and one corollary. The paper includes also four examples.

MSC:
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
14E20 Coverings in algebraic geometry
34M99 Ordinary differential equations in the complex domain
33E10 Lamé, Mathieu, and spheroidal wave functions
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