Re: LaguerreL[n,a,x] with negative n?

• To: mathgroup at smc.vnet.net
• Subject: [mg117756] Re: LaguerreL[n,a,x] with negative n?
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Thu, 31 Mar 2011 03:57:46 -0500 (EST)

See http://reference.wolfram.com/mathematica/tutorial/SpecialFunctions.html

"Special functions in Mathematica can usually be evaluated for arbitrary complex values of their arguments. Often, however, the defining relations given in this tutorial apply only for some special choices of arguments. In these cases, the full function corresponds to a suitable extension or analytic continuation of these defining relations. Thus, for example, integral representations of functions are valid only when the integral exists, but the functions themselves can usually be defined elsewhere by analytic continuation."

"When you use the function LegendreP[n,x] with an integer n, you get a Legendre polynomial. If you take n to be an arbitrary complex number, you get, in general, a Legendre function.

In the same way, you can use the functions GegenbauerC and so on with arbitrary complex indices to get Gegenbauer functions, Chebyshev functions, Hermite functions, Jacobi functions and Laguerre functions. Unlike for associated Legendre functions, however, there is no need to distinguish different types in such cases."

Bob Hanlon

---- Ben Whale <bwhale at maths.otago.ac.nz> wrote:

=============
Apologies if this has been answered else where.

How does Mathematica compute the value of LaguerreL[n,a,x] with negative
n? How is the function defined?

From what I understand n has to be positive, and in my (admittedly not
exhaustive) searching I only come across information that says that n
must be positive.

Cheers,
Ben

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