Background


I'm modeling Genetic Algorithm(GA) with Markov chains and deriving the expression for the expectation of the first hittig time in the MC with 1 absorbing state and $l-1$ transient states. This results is an expression for a sum involving square of a binomial coefficient


Problem
I need to find a closed expression for
$$\sum\_{k=0}^{l/2} \binom{l/2}{k}^2 p^{2k}$$


where $p$ is a function of $l$ and lies between 0 and 1.


So far I've found a closed expression for 
$$\sum\_{k=0}^{n}k^2 \binom{n}{k}^2$$


Any suggestions are very much appreciated.



Square of Binomial Coefficient
According to Mathematica, your sum equals:


$(1-p^2)^{l/2} \mbox{LegendreP}\left(l/2, \frac{1+p^2}{1-p^2}\right),$


or


$\, \_2F\_1\left(-\frac{l}{2},-\frac{l}{2};1;p^2\right)$


The second sum is 
$n^2 \binom{2 n-2}{n-1}.$


Ain't technology grand...


**EDIT** The real question is: why do you want to know? The expressions I give above allow you to get asymptotics, get ODE satisfied by the functions, etc, etc. If you want an expression in terms of "elementary functions" (whatever that means in this case), with very high probability there are not any (this is less certain here because these are *definite* summmations). I strongly advise you to read Petkovsek and Zeilberger's "A=B."


