0
$\begingroup$

Could someone help me find a tight upper bound for the following: $\sum\limits_{i=0}^{k/2-1} {n/4 \choose i}{3n/4 \choose k-i}$

It's essentially a Chu-Vandermonde summation of only the first $k/2$ terms so it's clearly upper bounded by ${n \choose k}$, but I'd like to know how fast it approaches ${n \choose k}$ as a function of k.

Thanks.

$\endgroup$
0
$\begingroup$

mathematica says this here $$\frac{\Gamma (n+1)}{\Gamma (k+1) \Gamma (-k+n+1)}-\binom{\frac{n}{4}}{\frac{k}{2}} \binom{\frac{3 n}{4}}{\frac{k}{2}} \, _3F_2\left(1,-\frac{k}{2},\frac{k}{2}-\frac{n}{4};\frac{k}{2}+1,-\frac{k}{2}+ \frac{3 n}{4}+1;1\right)$$ which containes a hypergeometric function

$\endgroup$
  • $\begingroup$ I was hoping for something simpler :) $\endgroup$ – qpip1982 Oct 5 '16 at 14:12
  • $\begingroup$ i someone has a simplier result? $\endgroup$ – Dr. Sonnhard Graubner Oct 5 '16 at 14:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.