Difference between revisions of "Pochhammer equation"
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− | + | 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 [[#References|[1]]] and C. Jordan [[#References|[2]]]. | ||
The Pochhammer equation has been integrated using the [[Euler transformation|Euler transformation]], and its particular integrals have the form | The Pochhammer equation has been integrated using the [[Euler transformation|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 | + | 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 | + | exactly $ m $ |
+ | of these solutions are linearly independent. To construct the other solutions other contours are used, including non-closed ones (see [[#References|[3]]], [[#References|[4]]]). The [[Monodromy group|monodromy group]] for the Pochhammer equation has been calculated (see [[#References|[3]]]). | ||
Particular cases of the Pochhammer equation are the Tissot equation (see [[#References|[4]]]), i.e. the Pochhammer equation in which | Particular cases of the Pochhammer equation are the Tissot equation (see [[#References|[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|Papperitz equation]]. | and the [[Papperitz equation|Papperitz equation]]. |
Latest revision as of 17:43, 16 December 2020
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) |
Pochhammer equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pochhammer_equation&oldid=14095