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 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? Gottfried Ultimate Fellow Posts: 766 Threads: 119 Joined: Aug 2007 09/11/2013, 06:30 PM (This post was last modified: 09/12/2013, 07:25 AM by Gottfried.) Hi Mike - (09/11/2013, 10:49 AM)mike3 Wrote: Letting $P_n(x) = \sinh((2n+1) \mathrm{arsinh}(x))$ (so your $f_{2n+1}(x) = 2 P_n(x/2)$), I noticed $P_0(x) = x$ $P_1(x) = 4x^3 + 3x$ $P_2(x) = 16x^5 + 20x^3 + 5x$ $P_3(x) = 64x^7 + 112x^5 + 56x^3 + 7x$ $P_4(x) = 256x^9 + 576x^7 + 432x^5 + 120x^3 + 9x$ ... Now look at the Chebyshev polynomials $T_n(x)$ for odd $n$... - yes I've also put a question in MSE but retracted it because just after posting it I had found the entry in wikipedia... which is a really good one btw. That solved also the problem of the connection between the sinh/asinh cosh/acosh and sin/asin and cos/acos versions (cosh(x) = cos(ix) and the resulting change of signs in the polynomials). Well, this did not give some "usual,more common" simple function of which the asinh is the Schröder-function and only a family of polynomials instead (perhaps there might be some expression by elementary functions, anyway). But what this continuous iterable function $2\sinh(a^h \cdot \operatorname{asinh}(x/2))$ gives at least is then a fractional interpolation for the index of the Chebychev-polynomials, which in turn are specifically useful for polynomial interpolation... What will this give to us...? At the moment I've put it aside and I'll take a breath to look at it later again: what it has originally been for (for me in my notepad) in the bigger picture. Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by Gottfried - 09/10/2013, 12:23 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by tommy1729 - 09/10/2013, 09:50 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by tommy1729 - 09/10/2013, 10:06 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by mike3 - 09/11/2013, 10:49 AM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by Gottfried - 09/11/2013, 06:30 PM RE: 2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? - by Gottfried - 09/11/2013, 08:32 PM

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