Closed form solution for a differential-difference equation

Considering a differential – difference equation :

$$g (t + 1,s) = cosh (s) + frac {alpha} {t}sinh (s)frac {d} {ds} g (t,s)$$

with $g (t, s = 0) =1$, I am looking for a closed form for its solution.I have tried
Mathematica to solve it using iteration:

ClearAll[g, t, s, α]; 
g[t_Integer, s_] := g[t][s]; 
g[0] = Function[{s}, 0]; 
g[1] = Function[{s}, 1]; 
g[t_Integer][s_] := 
  With[{}, g[t] = Function[{y}, Evaluate[Expand[Cosh[y] + (α/(t - 1))*Sinh[y]*
          D[g[t - 1, y], y]]]]; g[t][s]]

which gives for instance:

g[3, s]= Cosh[s] + 1/2 α Sinh[s]^2

My question is "how can I use Mathematica to find a closed form for $g(t,s)$?"

P.S. In fact, I am looking for the generating function $sum_{t=1}^{infty}g[t,s]z^{-t}$.