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Is there are a term for a generalized exponential function? The series expansions of sine and cosine look very similar to the exponential function's series expansion

\begin{align} & \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\ & \sum_{n = 0}^{\infty} \frac{(-x)^n}{n!} \\ & \sum_{n = 0}^{\infty} \frac{x^{2n}}{(2n)!} & & \sum_{n = 0}^{\infty} \frac{x^{2n + 1}}{(2n + 1)!} & \\ & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{2n}}{(2n)!} & & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{2n + 1}}{(2n + 1)!} \\ & \sum_{n = 0}^{\infty} \frac{x^{3n}}{(3n)!} & & \sum_{n = 0}^{\infty} \frac{x^{3n + 1}}{(3n + 1)!} & & \sum_{n = 0}^{\infty} \frac{x^{3n + 2}}{(3n + 2)!} \\ & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{3n}}{(3n)!} & & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{3n + 1}}{(3n + 1)!} & & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{3n + 2}}{(3n + 2)!} \\ & \vdots && \vdots && \vdots \end{align}

and so I was wondering as to whether or not there is a name for all of these types of infinite sums and what properties they all share.

William Ryman
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Denote by $\omega$ $k$-th root of unity. Then $$\sum_{n=0}^{\infty}\frac{x^{kn}}{(kn)!} = \frac{1}{k} \sum_{r = 1}^{k} e^{\omega^{r} x},$$ and hence there is not much need for the name. You can obtain combinatorial identities and asymptotic from simple analysis of $e^{zx}, z\in \mathbb{C}$.

Rybin Dmitry
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  • Thank you. How did you arrive at this? I know it must be obvious to you, but a derivation would be much appreciated. – William Ryman Jun 29 '22 at 04:47
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    This question is a duplicate. Please post your answer to the original question instead (if it differs from those given there). – Gary Jun 29 '22 at 05:25
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    @WilliamRyman Let $1, \omega, \omega^{2},..., \omega^{k - 1}$ denote all $k$-th roots of unity. Then the sum of $d$-th powers $1^{d} + w^{d} + w^{2d} + ... + w^{(k - 1)d}$ is equal to $0$ if $d$ is not divisible by $k$ and equal to $k$ if $d$ is divisible by $k$ (think of geometric proof in terms of rotations of regular $k$-gons). Hence whenever we want to filter out each $k$-th term in Taylor series, we use this summation over roots of unity, and put a factor $1/k$ in front. – Rybin Dmitry Jun 29 '22 at 05:25
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These are called Olivier functions. See, for example, L. Carlitz, Some arithmetic properties of the Olivier functions, Math. Ann. 128 (1955), 412–419.

Ira Gessel
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