eqL=D[P[n,t],t]

eqR=-(n+c)P[n,t]+(P[n+1,t](n+1)+c P[n-1,t])Exp[-s]

PP[n_,t_]=Exp[t ψs] cs^n/n!

(eqL/.P->PP)/PP[n,t]

Collect[FullSimplify[(eqR/.P->PP)/PP[n,t]],n,Simplify]

solcs=Solve[-1+((c (E^(-s)))/cs)==0,cs][[1]]

ψ[s_]=Simplify[-c+cs (E^(-s))/.solcs]

-D[ψ[s],s]

-ψ'[0]

Plot[-ψ'[s]/.c->1/4,{s,-1,2}]

Plot[ψ[s]/.c->1/4,{s,-1,2}]

sstar[k_]=s/.Solve[ψ'[s]==-k,s][[1]]/.C[1]->0

pi[k_]=ψ[sstar[k]]+sstar[k] k

Plot[pi[k]/.c->3/4,{k,0,4}]

pi'[2c]