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Posted 2018-12-03, 09:25 PM
in reply to Chruser's post "Maclaurin series similar to those of e^x"
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I was playing around with functions of the form }(x) = \sum_{n=0}^{\infty} \frac{x^{g(n)}}{(g(n))!}) (based on the trivial Maclaurin expansion of ) ), and noticed, for example, that
=\cosh(x),)
=\sinh(x),)
=\cosh(x)-1,)
=\sinh(x)-x,)
=\frac{\cosh(x) + \cos(x)}{2},)
=\frac{\sinh(x) + \sin(x)}{2},)
=\frac{\cosh(x) - \cos(x)}{2},)
=\frac{\sinh(x) - \sin(x)}{2},)
=\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  is the golden ratio. If you're sufficiently bored, you should help me find other, interesting functions ) , 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 }(x) = f_{g(n)-1}(x)) if you throw away any terms such that  = 0) .)
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Minor quibble, but shouldn't }(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 09:49 PM.
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