I was studying the function

finding that for odd a this gives (finite) polynomials in x with integer coeffcients - thus for integer x and odd a this is integer and for rational x this is also rational. (For even a replace sinh by cosh). So this is somehow interesting.

Formally this looks like the Schroeder-function of some unknown function (with some parameter a), where also the iteration-height h is introduced:

Is this function something common in our usual, daily toolbox?

finding that for odd a this gives (finite) polynomials in x with integer coeffcients - thus for integer x and odd a this is integer and for rational x this is also rational. (For even a replace sinh by cosh). So this is somehow interesting.

Formally this looks like the Schroeder-function of some unknown function (with some parameter a), where also the iteration-height h is introduced:

Is this function something common in our usual, daily toolbox?

Gottfried Helms, Kassel