I was studying the function
\( f_a(x) = 2 \cdot \sinh ( a \cdot \sinh^{-1}(x/2)) \)
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 \( f_a(x) \) (with some parameter a), where also the iteration-height h is introduced:
\( f_a^{[h]}(x) = 2 \cdot \sinh( a^h \cdot \sinh^{-1}(x/2)) \)
Is this function \( f_a(x) \) something common in our usual, daily toolbox?
\( f_a(x) = 2 \cdot \sinh ( a \cdot \sinh^{-1}(x/2)) \)
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 \( f_a(x) \) (with some parameter a), where also the iteration-height h is introduced:
\( f_a^{[h]}(x) = 2 \cdot \sinh( a^h \cdot \sinh^{-1}(x/2)) \)
Is this function \( f_a(x) \) something common in our usual, daily toolbox?
Gottfried Helms, Kassel