My goal is to solve for $L$ in
$\frac{(2k)!}{2^kk!}{2nL - L^2 \choose 2k} = \sum_{s=0}^k{L \choose s}{n-L \choose s}s!\frac{(2k-2s)!}{2^{k-s}(k-s)!)}{L-s \choose 2k-2s}.$
I tried to use the solve command in Maple to do this. I entered
solve(factorial(2*k)/(2^k*factorial(k)) = sum(binomial(l, s)*binomial(n-l, s)*binomial(l-s, 2*k-2*s)*factorial(s)*factorial(2*k-2*s)/(2^(k-s)*factorial(k-s)), s = 0 .. k), l)
And the output was
RootOf(binomial(_Z, 2*k)*hypergeom([-k, -n+_Z], [_Z-2*k+1], 2)-1)
I believe I have figured out that "_Z" is just a dummy variable. My first question is, why is there a variable at all? I'll note that I have little understanding of the hypergeometric function. But other things that I'd be interested in:
I'd be mostly happy to just have an approximate solution just in terms of elementary functions. Or to know the asymptotics. Can any of this be easily deduced?
Thanks!
_Z is a placeholder. What Maple is telling you is that the answer is some $z$ such that ${z \choose 2k}\; hypergeom([-k,z-n],[z-2k+1],2)-1 = 0$. Basically it "simplified" your sum into a hypergeometric function, cancelled some obvious factors, and then threw up its hands and said it doesn't know how to solve this equation. Well, neither do we. What is worse is that Maple doesn't know much about asymptotics of hypergeometric functions. So asymptotic or approximate solutions are not going to be easy to obtain.