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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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This requires
for . Therefore,
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for , 2, ..., so
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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