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 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 (so your ), I noticed

...

Now look at the Chebyshev polynomials for odd ...

- 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 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