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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.

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The general closed form is a sum of linear combinations of terms of the form e^{zeta x} where zeta is an nth root of unity. One way to see this is that f = exP_{m,n} satisfies the differential equation f^{(n)} = f. If you'd like to look up exactly what this linear combination is, the keyword is "discrete Fourier transform."

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This can be done starting with any power series and is called series multisection.

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The differential equation given by Qiaochu f(n) = f can be solved by means of the Laplace transform. We obtain:

snF(s)-s(n-m-1) = F(s)

The second term originates from the unvanishing m-th derivative of f(x). All other derivatives up to (n-1) vanish.

Applying the inverse Laplace transform, one finds that the coefficient of exp(ωix) in the expansion is:

ωi(-m-1)/i, where ωi is the i-th n-th root of unity

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Ricatti seems to have studied the "generalized hyperbolic functions" way way back, and there's also the Mittag-Leffler function (your original function is expressible in terms of Mittag-Leffler) that frequently crops up in the fractional calculus. You might want to try searching for papers on these things.

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