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Re: sum of binomials .. bug ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg70503] Re: [mg70486] sum of binomials .. bug ?
*From*: Devendra Kapadia <dkapadia at wolfram.com>
*Date*: Wed, 18 Oct 2006 04:16:45 -0400 (EDT)
*References*: <200610170659.CAA02125@smc.vnet.net>
On Tue, 17 Oct 2006, yann_che2 at yahoo.fr wrote:
> Hi everyone,
>
> on Mathematica 5.2 (mac os x), experimenting sums of binomials, i tried
> the following:
>
> In[6]:= f[k_] := Sum[Binomial[21 - k, i], {i, 0, 10 - k}]
> In[7]:= x = 3; f[x]
> Out[7]:= 63004
> In[8]:= Clear[x] ; f[x] /. x -> 3
> Out[8]:= 262144
> In[9]:= Clear[x] ; f[x]
> Out[9]:= 2^(21-x)
>
>
> does anyone know why Out[7] and Out[8] give different results ? do you
> think it is a bug ? i searched everywhere in the forums but couldn't
> find anything that helped.
> do you have a clue ?
>
> yann
>
Hello Yann,
Thank you for reporting the inconsistent answers in the
binomial sum given above.
The answer 2^(21-x) for f[x] in your example is incorrect.
Sums of this type are often evaluated by rewriting them in terms
of infinite hypergeometric series. The incorrect answer occurs
during this conversion.
A partial workaround for the problem is to replace the upper
limit '10-k' by 'a-k' as shown below. Using this method, we
recover the missing Hypergeometric2F1 term in the answer in
Out[5].
=============================
In[1]:= $Version
Out[1]= 5.2 for Linux (June 27, 2005)
In[2]:= f1[k_] := Sum[Binomial[21 - k, i], {i, 0, a-k}]
In[3]:= x=3;a=10;f1[x]
Out[3]= 63004
In[4]:= Clear[x, a];f1[x]/.{x-> 3, a-> 10}
Out[4]= 63004
In[5]:= f1[x]//InputForm
Out[5]//InputForm=
2^(21 - x) - (Gamma[22 - x]*Hypergeometric2F1[1, -20 + a, 2 + a - x, -1])/
(Gamma[21 - a]*Gamma[2 + a - x])
================================
I apologize for the confusion caused by this problem.
Sincerely,
Devendra Kapadia,
Wolfram Research, Inc.
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