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2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ?
#4
Letting (so your ), I noticed






...

Now look at the Chebyshev polynomials for odd ...






...

So . Then,

.

This, I suppose, is as "close as we can get to something in the 'usual' toolbox", at least if your "usual" toolbox has enough to include well-known named sequences of polynomials like the Chebyshev polynomials.

Note also that



and, for ,



. Now



Taking and using the correspondence between hyperbolic and trigonometric functions gives . Also, for cosh we get . Thus the result above simplifies to (the drops out due to the evenness of cosh) and so we have a formal proof of the relation to the Chebyshev polynomials we just gave.
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RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by mike3 - 09/11/2013, 10:49 AM

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