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Re: A hypergeom evaluation

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On 05.01.2012 19:01, Peter Luschny wrote:
> On 5 Jan., 14:47, Axel Vogt<&nore...@axelvogt.de>  wrote:
>
>> http://www.wolframalpha.com/input/?i=HypergeometricPFQ%28{1%2C+1%2F2%2C+0}%2C{1%2C+0}%2Cx%29
>> That shows the bug - giving 1 for all x.
>
> Hi Axel,
>
> I get the impression there is more than a single bug lurking beneath
> all that.
>
> First of all here is the expression which does what I want:
>
> G := (n,m) ->  (m/n)*2^(n-m)*binomial(2*n-1-
> m,n-1)*hypergeom([1/2+1/2*m, m-n, 1/2*m],[m, 1+m-2*n],2)
> -1/2*binomial(-m+1+2*n,n)*binomial(n-2,n-1)*hypergeom([1,1,-1/2*m
> +3/2+n,-1/2*m+1+n],[-m+2+n, n+1,-n+2],2);
>
> for n from 1 to 5 do seq(round(evalf(G(n,m))),m=1..n) od;
> 1
> 3, 1
> 14, 6, 1
> 77, 37, 9, 1
> 462, 238, 69, 12, 1
>
> However, when I use 'simplify' (like Joe) instead of 'evalf' I get
>
> for n from 1 to 5 do seq(simplify(G(n,m)),m=1..n) od;
>
> 1
> -3 I, 1
> -14 I, -8 - 14 I, 1
> -77 I, -40 - 77 I, -60 - 93/2 I, 1
> -462 I, -224 - 462 I, -336 - 288 I, -336 - 114 I, 1
>
> Peter
>

Hm, in Maple 15 that code does not execute, G(2,1) gives me
   "Error, (in hypergeom/check_parameters) function doesn't exist,
   found the number 0 in the second list of parameters"

So I used

g := (n,m,x) -> (m/n)*2^(n-m)*binomial(2*n-1-m,n-1)*
     Hypergeom([1/2+1/2*m, m-n, 1/2*m],[m, 1+m-2*n],x)
     -1/2*binomial(-m+1+2*n,n)*binomial(n-2,n-1)*
     Hypergeom([1,1,-1/2*m+3/2+n,-1/2*m+1+n],[-m+2+n, n+1,-n+2],x);

Then it works and gives me:

for n from 1 to 5 do seq(round(evalf(g(n,m,2))),m=1..n) od;

                                   1
                                  3, 1
                                14, 6, 1
                              77, 37, 9, 1
                          462, 238, 69, 12, 1

for n from 1 to 5 do seq(simplify(value( g(n,m,2) )),m=1..n) od;

                                   1
                                  3, 1
                                14, 6, 1
                              77, 37, 9, 1
                          462, 238, 69, 12, 1

What I can imagine (you have an old version and Maple certainly
worked on that hypgeom part as well):

Usually pFq for p=q+1 (as in your case) has a branch cut on the
real axis, starting in x=1, so your x=2 is *in* the branch cut.

Now Maple extends into the branch cut _counterclockwise_ seen
from the branch point (which x=1), if I remember correctly.
And that means: from below, i.e. limit(..., 2 + I*y, y=0,right).

May be there happens some difference between the (old) numerical
way and the symbolic way.

You may try by looking at 2 +- I* 2^(-k), k = 8,9,10, ...

But that may only affect the imaginary part (example: g(2,1,x)).

However in Maple 15 it works (up to notations as above).

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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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