Maclaurin series similar to those of e^x

Profile · 05:21 PM

I was playing around with functions of the form $\reverse f_{g(n)}(x) = \sum_{n=0}^{\infty} \frac{x^{g(n)}}{(g(n))!}$ (based on the trivial Maclaurin expansion of $\reverse e^x = f_n(x)$ ), and noticed, for example, that

$\reverse f_{2n}(x)=\cosh(x),$
$\reverse f_{2n+1}(x)=\sinh(x),$
$\reverse f_{2n+2}(x)=\cosh(x)-1,$
$\reverse f_{2n+3}(x)=\sinh(x)-x,$
$\reverse f_{4n}(x)=\frac{\cosh(x) + \cos(x)}{2},$
$\reverse f_{4n+1}(x)=\frac{\sinh(x) + \sin(x)}{2},$
$\reverse f_{4n+2}(x)=\frac{\cosh(x) - \cos(x)}{2},$
$\reverse f_{4n+3}(x)=\frac{\sinh(x) - \sin(x)}{2},$
$\reverse f_{5n}(x)=\frac{e^x}{5} + \frac{2}{5}\left(e^{-\varphi x/2}\cos\left(\frac{1}{2}\sqrt{\sqrt{5}\varphi^{-1}}x\right)+e^{\varphi^{-1}x/2} \cos\left(\frac{1}{2}\sqrt{\sqrt{5}\varphi}x\right )\right),$

where $\reverse \varphi$ is the golden ratio. If you're sufficiently bored, you should help me find other, interesting functions $\reverse g(n)$ , or similar series expansions, e.g. by using WolframAlpha. I suspect that the series above have been studied in some more general context since they're so "obvious", but I haven't seen anything along those lines previously.

(Edit: I should note that $\reverse f'_{g(n)}(x) = f_{g(n)-1}(x)$ if you throw away any terms such that $\reverse g(n) = 0$ .)

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