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

  • $\begingroup$ 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. $\endgroup$
    – Ferhat
    Jan 24, 2021 at 20:19


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