Often that notation can mean the second derivative of the function with respect to the first argument.
For example, in a fresh Wolfram|Alpha window I enter
D[ g[x,y],{x,2}]
which indicates taking the second derivative of g(x,y) with respect to x
then it promptly returns (approximately formatted here)
g^(2,0)(x,y)
If in a fresh Wolfram|Alpha window I enter
integral log(x)*2/x exp(-7x - 5/x) from 0 to inf
then it promptly returns
-2 log(7/5) K0(2 Sqrt(35))
and tells me that K0(x) is the modified Bessel function of the second kind.
