We do have elementary symmetric functions, SymmetricPolynomial[k,{x_1..x_n}]
http://reference.wolfram.com/mathematica/ref/SymmetricPolynomial.html

But I didn't found complete homogeneous symmetric functions.

The induction method to compute h_n from e_i and h_j (j<=n-1) is not that efficient. 
https://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomial

Is there any easier way to do this?