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)
How to Cite This Entry:
Pochhammer equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pochhammer_equation&oldid=50994
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article