Composition Kostka functions

Macdonald defined two-parameter Kostka functions K_{\lambda\mu}(q,t) where \lambda, \mu are partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture that also these more general Kostka functions are polynomials in q and t^{1/2} with non-negative integers as coefficients. If q=0 then our Kostka functions are Kazhdan-Lusztig polynomials of a special type. Therefore, our positivity conjecture combines Macdonald positivity and Kazhdan-Lusztig positivity and hints towards a connection between Macdonald and Kazhdan-Lusztig theory.