Pochhammer equation

A linear ordinary differential equation of order $ n $ of the form

$$ Q ( z) w ^ {( n)} - \mu Q ^ \prime ( z) w ^ {( n - 1 ) } + \dots + $$

$$ + ( - 1 ) ^ {n} \frac{\mu \dots ( \mu + n - 1 ) }{n!} Q ^ {( n)} ( z) w + $$

$$ - \Big [ R ( z) w ^ {( n - 1 ) } - ( \mu + 1 ) R ^ { \prime } ( z) w ^ {( n - 2 ) } + \dots + $$

$$ + ( - 1 ) ^ {( n - 1 ) } \frac{( \mu + 1 ) \dots ( \mu + n - 1) }{( n - 1 ) ! } R ^ {( n - 1 ) } ( z) w \Big ] = 0 , $$

where $ \mu $ is a complex constant and $ Q ( z) , R ( z) $ are polynomials of degree $ \leq n $ and $ \leq n - 1 $, 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

$$ \tag{* } w ( z) = \int\limits _ \gamma ( t - z ) ^ {\mu + n - 1 } u ( t) \ d t , $$

$$ u ( t) = \frac{1}{Q ( t) } \mathop{\rm exp} \left [ \int\limits ^ { t } \frac{ R ( \tau ) }{Q ( \tau ) } d \tau \right ] , $$

where $ \gamma $ is some contour in the complex $ t $- plane. Let all roots $ a _ {1} \dots a _ {m} $ of the polynomial $ Q ( z) $ be simple and let the residues of $ R ( z) / Q ( z) $ at these points be non-integers. Let $ a $ be a fixed point such that $ Q ( a) \neq 0 $ and let $ \gamma _ {j} $ be a simple closed curve with origin and end at $ a $, positively oriented and containing only the root $ a _ {j} $, $ j = 1 \dots m $, inside it. Formula (*) gives the solution of the Pochhammer equation, if with

$$ \gamma = \gamma _ {j} \gamma _ {k} \gamma _ {j} ^ {-1} \gamma _ {k} ^ {-1} ,\ \ j \neq k ,\ j , k = 1 \dots m , $$

exactly $ m $ 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

$$ Q ( z) = \prod _ {i = 1 } ^ { {n - 1} } ( z - a _ {j} ) ,\ \ R ( z) = Q ( z) \left ( 1 + \sum _ {j = 1 } ^ { {n - 1} } \frac{b _ j}{z - a _ {j} } \right ) , $$

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)