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Re: Hypergeometric functions and Sum error in 5.01?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51734] Re: [mg51678] Hypergeometric functions and Sum error in 5.01?
*From*: DrBob <drbob at bigfoot.com>
*Date*: Sun, 31 Oct 2004 01:16:27 -0500 (EST)
*References*: <200410290738.DAA03367@smc.vnet.net>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
It gets worse. Much worse.
This result is undefined for n==3 (for all m, presumably):
s1 = Sum[Binomial[n - i, m]*2^i, {i, 0, n - m}]
% /. n -> 3
(Gamma[1 + n]*Hypergeometric2F1[1, m - n, -n, 2])/(Gamma[1 + m]*Gamma[1 - m + n])
ComplexInfinity
That makes sense, as the Gamma function has nasty poles at zero and the negative integers.
Yet substituting 3 for n in the original expression gives (for all m, presumably):
Sum[Binomial[3 - i, m]*2^i, {i, 0, 3 - m}]
0
But substituting specific values for m, we have:
Sum[Binomial[3-i,1]2^i,{i,0,3-1}]
Sum[Binomial[3-i,2]2^i,{i,0,3-2}]
Sum[Binomial[3-i,3]2^i,{i,0,3-3}]
11
5
1
So we have three wildly different values for the same thing (for n==3 and a given value of m).
Looking at the definition of Binomial, of course we have:
Binomial[0, 3] == Gamma[0 + 1]/(Gamma[3 + 1]*Gamma[0 - 3 + 1])
True
But look at the second term in the denominator:
Gamma[0-3+1]
ComplexInfinity
Caveat emptor, Baby!
Bobby
On Fri, 29 Oct 2004 03:38:55 -0400 (EDT), Richard Ollerton <R.Ollerton at uws.edu.au> wrote:
> Mathematica 5.01 produces the following:
> In[1]:= s1=Sum[Binomial[n-i,m]2^i,{i,0,n-m}]
>
> Out[1]= Gamma[1+n] Hypergeometric2F1[1,m-n,-n,2] / (Gamma[1+m] Gamma[1-m+n])
>
> In[2]:= Table[{s1,Sum[Binomial[n-i,m]2^i,{i,0,n-m}]}/.n®3,{m,1,3}]
>
> Out[2]= {{-5,11},{-11,5},{-15,1}}
>
> Out[2] compares Mathematica's closed form with actual values. There appears to be an error in the rules used by Sum when simplifying this expression.
>Richard Ollerton
> r.ollerton at uws.edu.au
>
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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