All of the following outputs are wrong (in Mathematica 8.0), except for the last one where I plug in concrete values. Can anyone explain this to me? I consider this a rather serious issue because most people would probably rely on the output of Mathematica in any of the three cases when it comes to such an easy and well known identity (which is obviously not true for all integers).
In[6631]:= Reduce[Sum[Binomial[n, k] Binomial[s, t + k], {k, 0, n}] == Binomial[n + s, n + t], {n, s, t}, Integers]
Out[6631]=
n \[Element] Integers && s \[Element] Integers && t \[Element] Integers
In[6622]:= FindInstance[Sum[Binomial[n, k] Binomial[s, t + k], {k, 0, n}] != Binomial[n + s, n + t], {n, s, t}, Integers]
Out[6622]= {}
In[6632]:= ReplaceAll[Sum[Binomial[n, k] Binomial[s, t + k], {k, 0, n}] == Binomial[n + s, n + t], {n -> 1, s -> -1, t -> -1}]
Out[6632]= True
In[6623]:= n = 1;
s = -1;
t = -1;
Sum[Binomial[n, k] Binomial[s, t + k], {k, 0, n}]
Binomial[n + s, n + t]
Out[6626]= 2
Out[6627]= 1
