Hi Gottfried
Once again we see this 2sinh function !!
Im thinking about this function too, together with my friend mick.
I assume you know very well that when given \( f(x) \) we can compute that it is the superfunction of \( g(x) \) where \( g(x)=f(f^{-1}(x)+1) \). Although that formula does not simplify the expression if that is even possible.
There are many named (odd) polynomials in math and I am reminded that the similar looking (problem) \( sin (n arcsin(x)) \) and related ones are connected to ChebyshevPolynomials (of the First or Second Kind for example) and other named polynomials that are usually defined by differential equations.
So I guess we could say it is an iteration of the g-formula above for some named polynomials.
But I doubt that answers your questions.
Notice that - just like the Riemann hypothesis - the zero's of \( 2sinh((2n+1)arcsinh(z/2)) = 0 \) All lie on a critical line !!
Hint: \( cos(pi*m/(4n+2)) i \) !!
Im not sure how to answer your question more.
I guess we need to look at some named polynomials and their recursions.
Further just like the addition formula for sine was usefull , the same is probably true for this \( 2sinh \) !!
Btw did you read my fermat superfunction post ?
Elementary functions are still underrated !
regards
tommy1729
Once again we see this 2sinh function !!
Im thinking about this function too, together with my friend mick.
I assume you know very well that when given \( f(x) \) we can compute that it is the superfunction of \( g(x) \) where \( g(x)=f(f^{-1}(x)+1) \). Although that formula does not simplify the expression if that is even possible.
There are many named (odd) polynomials in math and I am reminded that the similar looking (problem) \( sin (n arcsin(x)) \) and related ones are connected to ChebyshevPolynomials (of the First or Second Kind for example) and other named polynomials that are usually defined by differential equations.
So I guess we could say it is an iteration of the g-formula above for some named polynomials.
But I doubt that answers your questions.
Notice that - just like the Riemann hypothesis - the zero's of \( 2sinh((2n+1)arcsinh(z/2)) = 0 \) All lie on a critical line !!
Hint: \( cos(pi*m/(4n+2)) i \) !!
Im not sure how to answer your question more.
I guess we need to look at some named polynomials and their recursions.
Further just like the addition formula for sine was usefull , the same is probably true for this \( 2sinh \) !!
Btw did you read my fermat superfunction post ?
Elementary functions are still underrated !
regards
tommy1729