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A hypergeom evaluation

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  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

#276 — A hypergeom evaluation

F := (n,m) -> (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*hypergeom([1/2+m/2,m/
2,m-n],[m,1+m-2*n],2);
round(evalf(F(1,1),64));

Maple VR5 gives the value -I. Is this correct?

Cheers, Peter

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#277

Peter Luschny <peter.luschny@googlemail.com> writes:

> F := (n,m) -> (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*hypergeom([1/2+m/2,m/
> 2,m-n],[m,1+m-2*n],2);
> round(evalf(F(1,1),64));
>
> Maple VR5 gives the value -I. Is this correct?
>
> Cheers, Peter

with Maple 15:

simplify(F(1,1));
                    -I

-- 
Joe Riel

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#278

On 04.01.2012 19:42, Joe Riel wrote:
> Peter Luschny<peter.luschny@googlemail.com>  writes:
>
>> F := (n,m) ->  (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*hypergeom([1/2+m/2,m/
>> 2,m-n],[m,1+m-2*n],2);
>> round(evalf(F(1,1),64));
>>
>> Maple VR5 gives the value -I. Is this correct?
>>
>> Cheers, Peter
>
> with Maple 15:
>
> simplify(F(1,1));
>                      -I
>

Yes, it simplifies to 1/sqrt( 1 - z ) and z=2 gives it.

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#280

On 4 Jan., 19:42, Joe Riel <j...@san.rr.com> wrote:
> Peter Luschny <peter.lusc...@googlemail.com> writes:

> > F := (n,m) -> (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*hypergeom([1/2+m/2,m/
> > 2,m-n],[m,1+m-2*n],2);
> > round(evalf(F(1,1),64));
> > Maple VR5 gives the value -I. Is this correct?

> with Maple 15:
> simplify(F(1,1));
                     -I

Thank you Joe! Wolfram Alpha computes 1.

Table[(m/n) 2^(n - m) Binomial[2 n - m - 1, n - m]
HypergeometricPFQ[{1/2 + m/2, m/2, m - n}, {m, 1 + m - 2 n}, 2], {n,
4}, {m, n}]
{{1}, {3, 1}, {14, 6, 1}, {77, 37, 9, 1}}

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#282

In article
<6d7f5214-ccd9-4a00-aaa2-4d6734cffb03@v14g2000yqh.googlegroups.com>,
Peter Luschny <peter.luschny@googlemail.com> wrote:

> On 4 Jan., 19:42, Joe Riel <j...@san.rr.com> wrote:
> > Peter Luschny <peter.lusc...@googlemail.com> writes:
> 
> > > F := (n,m) -> (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*hypergeom([1/2+m/2,m/
> > > 2,m-n],[m,1+m-2*n],2);
> > > round(evalf(F(1,1),64));
> > > Maple VR5 gives the value -I. Is this correct?
> 
> > with Maple 15:
> > simplify(F(1,1));
>                      -I
> 
> Thank you Joe! Wolfram Alpha computes 1.
> 
> Table[(m/n) 2^(n - m) Binomial[2 n - m - 1, n - m]
> HypergeometricPFQ[{1/2 + m/2, m/2, m - n}, {m, 1 + m - 2 n}, 2], {n,
> 4}, {m, n}]
> {{1}, {3, 1}, {14, 6, 1}, {77, 37, 9, 1}}
> 

Since 2 is outside the radius of convergence, you will have to read the
documentation to find out what each of these systems does.

-- 
G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/

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#283

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.

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#284

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

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#285

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.
...
> Peter

I meant, that Mathematica has a bug, since in the unit
disc that stands for 1/sqrt(1-x).

Sorry for the odd link, HypergeometricPFQ([1, 1/2, 0], [1, 0], x)
is what you have to enter at Wolframalpha, it is f(1,1, x) where

f:= (n,m, x) -> (m/n)*2^(n-m)*binomial(2*n-m-1,n-m)*
   Hypergeom([1/2+m/2,m/2,m-n],[m,1+m-2*n],x); # Upper Case !

Maple writes it as Hypergeom([1, 1/2, 0],[1, 0],x). Either using
'value' or considering that in the unit disc as series it cab be
written as hypergeom([1/2],[],x), since the same nominators and
denominators cancel out.

Using simplify (or looking up books) this can be seen as square
root.

For the other part in your post I will answer separately.

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#287

> 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)?

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#290

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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#291

Axel, I agree with most of what you said.

>   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,

The question here is: What is the reason that Maple V choose (2) over
(3)? Is this justified? The problem as stated -- H(n,m,x) -- does not
indicate that one parameter is to preferred over the other.

I am tending towards the answer (1). By definition the hypergeometric
sum 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);

and this function is not well defined at n=1,m=1. Thus what you
reported

>> 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"

looks reasonable for me (and makes the answer of Mathematica
incomprehensible).

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#292

On 06.01.2012 12:51, Peter Luschny wrote:
> Axel, I agree with most of what you said.
...
> The question here is: What is the reason that Maple V choose (2) over
> (3)? Is this justified? The problem as stated -- H(n,m,x) -- does not
> indicate that one parameter is to preferred over the other.
....

I do not know, but these are different ways

(3*) limit(hypergeom([1/2, 1-n, 1],[1, 2-2*n],2),n = 1);
      evalf[10](%);

Even if I do not know, what Maple does in evalf/limit to accelerate
it will certainly evaluate n close (but not identical) to 1, which
is ~ 0.5 + 0.5*I, thus the answer.

(2*) hypergeom([m-n, 1/2*m, 1/2+1/2*m],[m, 1+m-2*n],x)
      eval(%, n=1);
      simplify(%);subs(x=2,%);
      eval(%, m=1);

through the 2nd command 'cancels' identical terms in the first and
the second parameter list, a kind of 'lifting away the problem',
the simplify converts it to a square root.

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#294

In article <9moccvF155U1@mid.individual.net>, Axel Vogt
<&noreply@axelvogt.de> wrote:

> On 06.01.2012 12:51, Peter Luschny wrote:
> > Axel, I agree with most of what you said.
> ...
> > The question here is: What is the reason that Maple V choose (2) over
> > (3)? Is this justified? The problem as stated -- H(n,m,x) -- does not
> > indicate that one parameter is to preferred over the other.
> ....
> 
> I do not know, but these are different ways
> 
> (3*) limit(hypergeom([1/2, 1-n, 1],[1, 2-2*n],2),n = 1);
>       evalf[10](%);
> 
> Even if I do not know, what Maple does in evalf/limit to accelerate
> it will certainly evaluate n close (but not identical) to 1, which
> is ~ 0.5 + 0.5*I, thus the answer.
> 
> (2*) hypergeom([m-n, 1/2*m, 1/2+1/2*m],[m, 1+m-2*n],x)
>       eval(%, n=1);
>       simplify(%);subs(x=2,%);
>       eval(%, m=1);
> 
> through the 2nd command 'cancels' identical terms in the first and
> the second parameter list, a kind of 'lifting away the problem',
> the simplify converts it to a square root.
> 
> 
> 
> 
> 

I do recall that a hypergeom need not be continuous (as a function of
the parameters) when a parameter passes an integer.

-- 
G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/

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#286

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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#288

On 5 Jan., 20:05, Axel Vogt <&nore...@axelvogt.de> wrote:

> 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"

Makes sense according to the first case in my last message.

> 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);

Why Hypergeom and not hypergeom?

> Then it works and gives me:
> for n from 1 to 5 do seq(round(evalf(g(n,m,2))),m=1..n) od;

Ok, so in this case there is no difference to my results.

> for n from 1 to 5 do seq(simplify(value( g(n,m,2) )),m=1..n) od;
> However in Maple 15 it works (up to notations as above).

So they fixed at least one bug in the last 10 releases :)
But why does my version G(2,1) not work any more?

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#289

On 05.01.2012 21:17, Peter Luschny wrote:
> On 5 Jan., 20:05, Axel Vogt<&nore...@axelvogt.de>  wrote:
>
>> 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"
>
> Makes sense according to the first case in my last message.
>
>> 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);
>
> Why Hypergeom and not hypergeom?

It does not evaluate ... Like "Int" and "int". Used some HYP
as dummy function before to see what happens to binomial.
A kind of reflex to get rid of the specific situation (which
once I was educated to do).

>
>> Then it works and gives me:
>> for n from 1 to 5 do seq(round(evalf(g(n,m,2))),m=1..n) od;
>
> Ok, so in this case there is no difference to my results.
>
>> for n from 1 to 5 do seq(simplify(value( g(n,m,2) )),m=1..n) od;
>> However in Maple 15 it works (up to notations as above).
>
> So they fixed at least one bug in the last 10 releases :)
> But why does my version G(2,1) not work any more?

Do not know, can not think 10 releases back :-)

Will (try to) answer the other post tomorrow ...

BTW: Gutes Neues! Axel

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