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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
##### Keywords:
Lamé operators; prescribed monodromy; monodromy group
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