Today, during the evening chat, Emmanuel Abbe threw an interesting question: Whether the sum of square roots of consecutive binomial coefficients converge to some closed form! That is, . We tried a few known combinatorics tweak, but no meaningful solution arrived. We were also wondering whether this has some asymptotic limit, but that too did not yield anything. A quick check on Mathematica wasn’t helpful either. Now the question is: Does this sum yield some closed form expression.
While playing with this sum in Mathematica, I found that for the sum of squares of binomial coefficients, there is a nice simple closed form.
I was toying with a proof. It turns out that, the proof is extremely simple and is a one line tweak of the Vandermonde identity . Simply substitute and we have the results on table. The natural question then would be: Is there a generalization for for any . Ofcourse now for it is trivial.
Apparently, it turns out that, there is no closed form expression for a general (all) . There are some interesting divisibility properties of these sums. An interesting account of that is addressed by Neil Calkin (Factors of sums of powers of binomial coefficients).
At the moment, I get a feeling that sum of fractional powers of binomial coefficients is not trivial. May be there is no closed form. May be not!