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Groups > comp.soft-sys.math.maple > #290

Re: A hypergeom evaluation

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On 05.01.2012 20:57, Peter Luschny wrote:
>> On 5 Jan., 19:44, Axel Vogt<&nore...@axelvogt.de>  wrote:
>> I meant, that Mathematica has a bug, since in the unit
>> disc that stands for 1/sqrt(1-x).
>
> Yes, I understood this. So let us look at the culprit
> hypergeom([1/2+m/2,m/2,m-n],[m,1+m-2*n],2);
>
> According to the definition this is
> H := (n,m,x) ->  sum(pochhammer(1/2+m/2,k)*pochhammer(m/
> 2,k)*pochhammer(m-n,k)/(pochhammer(m,k)*pochhammer(1+m-2*n,k))*(x^k/
> k!),k=0..infinity);
>
> Let us evaluate H(1,1,x) in three different ways.
>
> (1) H(n,1,x); simplify(%);subs(x=2,%); gives
> hypergeom([1/2, 1-n],[2-2*n],2)
>
> At n=1 the lower parameter is zero and thus the function is not well
> defined at n=1.
>
> (2) H(1,m,x); simplify(%);subs(x=2,%); gives
> (1/2-1/2*I)*(1/2+1/2*I)^(-m)
> which is -I at m=1. This is the value Maple returns.
>
> (3) limit(H(n,m,x),n=1);subs(x=2,%);subs(m=1,%);evalf(%);
> .5000000000-.5000000000*I
>
> Why is (2) 'more correct' :) than (1) or (3)?

Your H(n,m,x) is the Gaussian series 3F2(a,b,c, x) and H(n,1,x) is some
2F1(a,b,c, x), where I re-write it using Upper Case Sum (avoiding evaluation)
http://en.wikipedia.org/wiki/Gaussian_hypergeometric_series

A power series converges in a disc around x=0 within it radius of convergence r.
In general one has r=1 and it does not converge on the closed disc. However in
special cases depending on parameters it converges in x=+1 (Gauss).

So one can not simply insert x=2 in general,

   H(n,1,x); convert(%, GAMMA);
   eval(%, n=4/3); eval(%, x=2); value(%);

will give infinity.

Now extend it as 2F1=hypergeom([1/2, 1-n],[2-2*n],x) from the unit disc
by using the 'value' command, the Maple will refuse to handle n=1.

This is your way (1).

However writing the series as H(n,1,x); convert(%, GAMMA); one sees,
that it has the term GAMMA(2-2*n)/GAMMA(1-n) and the limit in n=1 is
1/2, which Maple uses for the following:

   H(n,1,x); #convert(%, GAMMA);
   limit(%, n=1);
   value(%);

     1/2 * 1/sqrt(1-x)

This was your way (3). While you way (2) can be written as

   H(1,m,x); #convert(%, GAMMA);
   limit(%, m=1);
   value(%);

     1/sqrt(1-x)

Thus you are taking limits in the parameters in different orders,
I would say.

Note however, that for x=2 taking those limits is questionable

   r:=4;
   H(1 + 1/2^r,1, 2); value(%);
   H(1, 1 + 1/2^r, 2); value(%);

both are infinite.

Thus all that is usually understood as function in x, not in the
parameters, as it behave quite odd (guess that is one reason, why
Maple refuses its work for the way (1)).

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Thread

A hypergeom evaluation Peter Luschny <peter.luschny@googlemail.com> - 2012-01-04 09:22 -0800
  Re: A hypergeom evaluation Joe Riel <joer@san.rr.com> - 2012-01-04 10:42 -0800
    Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-04 19:59 +0100
    Re: A hypergeom evaluation Peter Luschny <peter.luschny@googlemail.com> - 2012-01-04 11:08 -0800
      Re: A hypergeom evaluation "G. A. Edgar" <edgar@math.ohio-state.edu.invalid> - 2012-01-05 06:21 -0700
        Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-05 14:47 +0100
          Re: A hypergeom evaluation Peter Luschny <peter.luschny@googlemail.com> - 2012-01-05 10:01 -0800
            Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-05 19:44 +0100
              Re: A hypergeom evaluation Peter Luschny <peter.luschny@googlemail.com> - 2012-01-05 11:57 -0800
                Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-06 11:37 +0100
                Re: A hypergeom evaluation Peter Luschny <peter.luschny@googlemail.com> - 2012-01-06 03:51 -0800
                Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-06 14:47 +0100
                Re: A hypergeom evaluation "G. A. Edgar" <edgar@math.ohio-state.edu.invalid> - 2012-01-07 06:10 -0700
            Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-05 20:05 +0100
              Re: A hypergeom evaluation Peter Luschny <peter.luschny@googlemail.com> - 2012-01-05 12:17 -0800
                Re: A hypergeom evaluation Axel Vogt <&noreply@axelvogt.de> - 2012-01-05 23:02 +0100

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