[SOLVED] Is there a way to specify integer arguments while integrating in Mathematica

I am trying to evaluate the following integral in Mathematica:

Integrate[u ^(m + 1)*Exp[-(u^2)/(2*rho)]*(u^2 - eps)^(j + k/2 - 1) , {u, Sqrt[eps],Infinity}]

I want to specify that for me, m and j are nonnegative integers, and k is a positive integer. Is there a way to do that in Mathematica? Thanks in advance!.

Thanks to @Bill's answer below, I was able to use:

Integrate[
 u ^(2*m + 1)*Exp[-(u^2)/(2*rho)]*(u^2 - eps)^(j + k/2 - 1) , {u, 
  Sqrt[eps], Infinity}, 
 Assumptions -> 
  m >= 0 && j >= 0 && k > 0 && Element[m | j | k, Integers] ]

However, the answer I get: involves Gamma(-m) when I use 2*m as an even integer (m here is an integer) which does not exist.