Let a generalised exponential function exp_{m,n} (I'm not sure if this notation is already used by anything else) be defined as such, for n a positive integer and m between 0 and n-1 (inclusive):

exp_{m,n} (x) = sum of (x^(nk+m)/(nk+m)!) as k ranges from 0 to infinity.

(e.g., cosh = exp_{0,2}, sinh = exp_{1,2}.) What are the closed forms for such generalised exponentials (for arbitrary n) in terms of real transformations of the elementary functions? Using e^(ωx) (where ω is a primitive cube root of 1) a closed form for exp_{1,3} - exp_{2,3} can be obtained, but I'm not sure about the individual functions.