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Re: Re: Jacobi polynomials in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg99086] Re: [mg99063] Re: Jacobi polynomials in Mathematica
  • From: danl at wolfram.com
  • Date: Sun, 26 Apr 2009 01:40:40 -0400 (EDT)
  • References: <gsrqsm$rac$1@smc.vnet.net> <gss3d4$2po$1@smc.vnet.net>

> On 24 Apr, 11:13, Jens-Peer Kuska <ku... at informatik.uni-leipzig.de>
> wrote:
>> Hi,
>>
>> a simple check
>>
>> JacobiP[n, a, b, z] /. {n -> 0.5, a -> 0.5, b -> 0.5, z -> 0.2}
>>
>> gives
>>
>> 0.767081
>>
>> so, it may be defined.
>>
>> Regards
>>    Jens
>>
>> Cora L wrote:
>> > Hello,
>> > I have a simple question: in Mathematica the Jacobi polynomials are
>> > implemented
>> > as JacobiP[n, a, b, z], see
>> >http://mathworld.wolfram.com/JacobiPolynomial.html
>>
>> > Is JacobiP[n, a, b, z] also defined if n is not an integer? More
>> > general, is
>> > JacobiP[n, a, b, z] defined for all real n, a, b and z?
>>
>> > Thanks!
>
> Well, even if Mathematica is giving out a value I'm not too sure
> whether it's correct or not.
>
> For example,
> Product[j, {j, 1, 4}] gives 24
>
> But
> Product[j, {j, 1, 4.5}] also gives 24.
>
> Surely, the second answer is wrong.

Not at all wrong. It is very much in accordance with documented behavior
for Product and iterators in Mathematica. Checking the relevant
documentation

tutorial/SomeGeneralNotationsAndConventions#27272

one sees

"...The iteration parameters Subscript[i, min],Subscript[i, max] and di do
not need to be integers. The variable i is given a sequence of values
starting at Subscript[i, min], and increasing in steps of di, stopping
when the next value of i would be greater than Subscript[i, max]. The
iteration parameters can be arbitrary symbolic expressions, so long as
(Subscript[i, max]-Subscript[i, min])/di is a number..."

I will remark that this is common behavior for iterators in other
languages as well.

Your example, using Product, is not an apt analogy to JacobiP. The reason
is that for specific iterator bounds (e.g. 1 to 4.5) it is reasonable to
expect behavior that is procedural (or at least algorithmic), rather than
some form of analytic result. A closer analogy to JacobiP might be the
relation of Factorial (which can be represented algorithmically via
Product) to Gamma.

To put it a bit differently, there is no reason to expect Product
evaluations in general to give closed-form functions that have "nice"
continuations. Closed for functions such as Jacobi polynomials, in
contrast, might have well established continuations to non-integer values.

One last remark: The statement "Well, even if..." is lacking in contest.
If you have no expectation of what JacobiP[noninteger,...] does or should
do, it is axiomatic that you do not know whether specific values are
"correct or not". Without some background information it is really unclear
what you are expecting from these evaluations.

Daniel Lichtblau
Wolfram Research




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