       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: Sat, 30 Oct 2004 03:49:09 -0400 (EDT)
• References: <200410290738.DAA03367@smc.vnet.net>
• 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:= s1=Sum[Binomial[n-i,m]2^i,{i,0,n-m}]
>
> Out= Gamma[1+n] Hypergeometric2F1[1,m-n,-n,2] / (Gamma[1+m] Gamma[1-m+n])
>
> In:= Table[{s1,Sum[Binomial[n-i,m]2^i,{i,0,n-m}]}/.n®3,{m,1,3}]
>
> Out= {{-5,11},{-11,5},{-15,1}}
>
> Out 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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