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Posted 2018-12-03, 10:25 PM in reply to Chruser's post "Maclaurin series similar to those of e^x"
Chruser said: [Goto]
 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 $\reverse g(n) = 0$.)
Minor quibble, but shouldn't $\reverse f'_{g(n)}(x)$ have a g(n) factor in each term? I haven't looked at this with pen and paper, so I might be missing some simplification, but that derivative formula looks wrong at first glance in the general case.

I'm also not sure I understand what you're looking for. Sin, sinh, etc are all essentially special functions, and you could always define special functions based on whatever series you come up with by replacing n with 2n or what have you. So are you essentially looking for series where such replacements produce things in terms of known special functions? Or am I misunderstanding?

Last edited by Demosthenes; 2018-12-03 at 10:49 PM.
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