# 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 $$\sum_{t=1}^{\infty}g[t,s]z^{-t}$$.

• I have found that the series $g(t,s)=\sum_{n=0}^{t-1}a[n,t]Cosh[n s]$ is the solution. However, finding the coefficients are yet hard to find. – Ferhat Jan 24 at 20:19