ahh, stupid....

Of course the basic Chebychev-polynomials are "the-functions-of which-acosh(x)-is-the-SchrĂ¶derfunction".

Let for instance then of course the iterates and the iterates to any height h are which can then be evaluated/interpolated using the cosh/acosh-pair as .

So my question is then -very simple- answered for "base=2" that the "more common" function "in our toolbox" is the quadratic polynomial (where the fractional iterates become power series instead...).

And if I take any higher-indexed Chebychev-polynomial as the base-function , the iterate is simply computable by the cosh/acosh-mechanism

(And for my initial question using sinh/asinh it is simply the same except with other polynomials)

Should have seen this before... -

Gottfried

[update]: And -oh wonder- this is just related to a question in MSE where I was involved this days without knowing that this two questions are in the same area; I just added the information about the cosh/arccosh-composition there... :-) http://math.stackexchange.com/questions/...965#490965

Of course the basic Chebychev-polynomials are "the-functions-of which-acosh(x)-is-the-SchrĂ¶derfunction".

Let for instance then of course the iterates and the iterates to any height h are which can then be evaluated/interpolated using the cosh/acosh-pair as .

So my question is then -very simple- answered for "base=2" that the "more common" function "in our toolbox" is the quadratic polynomial (where the fractional iterates become power series instead...).

And if I take any higher-indexed Chebychev-polynomial as the base-function , the iterate is simply computable by the cosh/acosh-mechanism

(And for my initial question using sinh/asinh it is simply the same except with other polynomials)

Should have seen this before... -

Gottfried

[update]: And -oh wonder- this is just related to a question in MSE where I was involved this days without knowing that this two questions are in the same area; I just added the information about the cosh/arccosh-composition there... :-) http://math.stackexchange.com/questions/...965#490965

Gottfried Helms, Kassel