Hyperbolic and recursion

[sinh(x) + cosh(x)]*[sinh(y) + cosh(y)] = sinh(x+y) + cosh(x+y)

But we can define some subset of hyperbolic angles such that:

r-1/r = a then we have a recursion, often trivial, of the form:
r^n = a*r^(n-1) + r^(n-2).  The recursion have the r = a/2+root(a^2+4)/2 and 1/r = -a/2 +root(a^2+4)/2

so, r-1/r = sinh(log(r)) =a and log(r) =  k is the base hyperbolic angle.
and sinh(k) = a and cosh(k) = root(a^2+4)  .

1/2(s(k) + c(k)) = r, and (s(k)+c(k))^n = [(1/2)*r]^n= s(nk)+c(nk)
1/2(s(k) - c(k)) = 1/r, and (s(k)+c(k))^n = [(1/2)/r]^n= s(nk)-c(nk)

I likely have messed up the factor of two.

sinh(k)^n =  a^n
cosh(k)^n =  root(a^2+4)^n
tanh(k)^n =a^n /root(a^2+4)^n

The recursions work fine, and easy to make but are discrete. I can  make a recursion in e.  All the derivatives are with respect to integer quants, and all the ratios go to rational fractions with an integer power. I can use tanh series to approximately solve differentials then let my ratio go to e in the limit.