Pochhammer equation

A linear ordinary differential equation of order of the form

where is a complex constant and are polynomials of degree and , respectively. The Pochhammer equation was studied by L. Pochhammer [1] and C. Jordan [2].

The Pochhammer equation has been integrated using the Euler transformation, and its particular integrals have the form

(*)

where is some contour in the complex -plane. Let all roots of the polynomial be simple and let the residues of at these points be non-integers. Let be a fixed point such that and let be a simple closed curve with origin and end at , positively oriented and containing only the root , , inside it. Formula (*) gives the solution of the Pochhammer equation, if with

exactly of these solutions are linearly independent. To construct the other solutions other contours are used, including non-closed ones (see [3], [4]). The monodromy group for the Pochhammer equation has been calculated (see [3]).

Particular cases of the Pochhammer equation are the Tissot equation (see [4]), i.e. the Pochhammer equation in which

and the Papperitz equation.

References

[1]	L. Pochhammer, "Ueber ein Integral mit doppeltem Umlauf" Math. Ann. , 35 (1889) pp. 470–494
[2]	C. Jordan, "Cours d'analyse" , 3 , Gauthier-Villars (1915)
[3]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
[4]	E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947)