Hi Mike -
- 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
(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