A binomial sum

[Henry Gould]

Eric W. Weisstein wrote:
Doesn't the following qualify?

Mathematica 5.0 for Mac OS X
Copyright 1988-2003 Wolfram Research, Inc.
 -- Terminal graphics initialized --

In[1]:= FullSimplify[Sum[(-1)^m Binomial[m + n, m], {m, 0, n}]]//InputForm

Out[1]//InputForm=
(2^(-n) + ((-1)^n*Gamma[3 + 2*n]*Hypergeometric2F1Regularized[1, 2 + 2*n,
     2 + n, -1])/Gamma[2 + n])/2

In[2]:= Table[{Sum[(-1)^m Binomial[m + n, m], {m, 0, n}],
        (2^(-n) + ((-1)^n*Gamma[3 + 2*n]*Hypergeometric2F1Regularized[1, 2 +
2*n,
     2 + n, -1])/Gamma[2 + n])/2},
        {n,0,10}]

Out[2]= {{1, 1}, {-1, -1}, {4, 4}, {-13, -13}, {46, 46}, {-166, -166},

>>    {610, 610}, {-2269, -2269}, {8518, 8518}, {-32206, -32206},
>>    {122464, 122464}}


In[3]:= Subtract@@@%

Out[3]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

Cheers,
-Eric

= = = = =
Well, Eric, if you call a hypergeometric function a "closed form", yes, but
that is no better than the original finite summation. It cannot be put as a
simple product and ratio of factorials.
Of course, even the function "n!" is NOT a  "closed form". But we accept
that as such.
The classical view is that the hypergeometric function is not a "closed
form". I would call that a  transformation of the original sum.
But then perhaps neither are sine, cosine, tangent and exp"closed forms".
As we say in West Virginia, it is all relative.
Sholom,
Henry