Re: sum of binomials .. bug ?

• To: mathgroup at smc.vnet.net
• Subject: [mg70503] Re: [mg70486] sum of binomials .. bug ?
• 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.

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,