Is there an analogue of the Fourier transform based on hyperbolic trig functions?

Question

Is there something analogous to Fourier series or the Fourier transform but which is based on hyperbolic trig functions rather than $\sin, \cos$, and $\exp$?

Answer 1

I think so.

$\ln|1+x|=x-x^2/2+x^3/3-x^4/4+...$

Let $e^y=1+x$.

Then $y=(e^y-1)-(e^y-1)^2/2+...$

and we know $e^{ky}=\cosh ky + \sinh ky$.

$(e^y-1)^n = \sum _{k=0}^n C(n,k)(\cosh ky + \sinh ky)(-1)^{n-k}$

$y= \sum_{n=1}^\infty \sum_{k=0}^nC(n,k)(\cosh ky +\sinh ky)(-1)^{k+1}/n $