Is there a formula for $k\pi ^n$, if $n$ is an odd number and $k$ is a rational number? I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number. 

I suspect the formula for $n$ even you are talking about comes from the riemann zeta function seen here. In the same articles you can read that a general formula for $\zeta(2k+1)$ is still an open problem.
