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The Laguerre differential equation is a special case of the more general ``associated Laguerre differential equation''
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with
. Note that if
, then the solution to the associated Laguerre differential equation is of the
form
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and the solution can be found using an Integrating Factor
so
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The associated Laguerre differential equation has a Regular Singular Point at 0 and an Irregular
Singularity at
. It can be solved using a series expansion,
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![\begin{displaymath}[(n+1)a_1+\lambda a_0]+ \sum_{n=1}^\infty \{[(n+1)n+(\nu+1)(n+1)]a_{n+1}-na_n+\lambda a_n\}x^n = 0
\end{displaymath}](l1_189.gif) |
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![\begin{displaymath}[(n+1)a_1+\lambda a_0]+ \sum_{n=1}^\infty [(n+1)(n+\nu+1)a_{n+1}+(\lambda-n)a_n]x^n = 0.
\end{displaymath}](l1_190.gif) |
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This requires
for
. Therefore,
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for
, 2, ..., so
![\begin{displaymath}
y=a_0\left[{1-{\lambda \over \nu+1}x-{\lambda (1-\lambda )\o...
...-\lambda )\over 2\cdot 3(\nu+1)(\nu+2)(\nu+3)}+\cdots}\right].
\end{displaymath}](l1_198.gif) |
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If
is a Positive Integer, then the series terminates and the solution is a Polynomial, known
as an associated Laguerre Polynomial (or, if
, simply a
Laguerre Polynomial).
See also Laguerre Polynomial
© 1996-9 Eric W. Weisstein
1999-05-26