       Re: Re: Jacobi polynomials in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg99083] Re: [mg99063] Re: Jacobi polynomials in Mathematica
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Sun, 26 Apr 2009 01:40:07 -0400 (EDT)
• References: <gsrqsm\$rac\$1@smc.vnet.net> <gss3d4\$2po\$1@smc.vnet.net>

```This is entirely correct:

Product[j, {j, 1, 4.5}]

24

...since j varies from the lower limit (1 in this case), adding 1 at each
step, and stopping at OR BELOW the upper limit. In this case j takes on
the values 1, 2, 3, and 4. Adding 1 again to get 5 would exceed the upper
limit.

But consider the calculations

Product[j, {j, 1, n}]
% /. n -> 4.5
Gamma[4.5 + 1]

n!

52.3428

52.3428

The first result generalizes the Product to an arbitrary positive integer
and returns Factorial, an INCORRECT result for non-integers such as 4.5.

Yet it's a USEFUL result for non-integers, if we intend to use the Gamma
function to extend Factorial.

Bobby

On Sat, 25 Apr 2009 03:52:46 -0500, Cora L <cora.lahnstein at googlemail.com>
wrote:

> 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.
>

--
DrMajorBob at bigfoot.com

```

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