I tried to define a simple rule defining how λ acts on
myrule1 = λ ψ[n_] -> α[n + 1] ψ[n + 1];
The result I get is correct provided there's just one λ on the RHS of
ψ[n]. For instance:
λ^2 ψ[n] //. myrule1
isn't computed at all. On the other hand, if I do it step by step:
λ α[1 + n] ψ[1 + n] /. myrule1
I get the correct result. I tried to define a new rule:
myrule2 = λ^m_ ψ[n_] -> α[n + 1] λ^(m - 1) ψ[n + 1];
but it doesn't work.
Since the recursive method seemed to work, I created a function which multiplies
ψ[m] by λ
timesλ[n_] := Nest[Times[λ, #] /. myrule1 &, ψ[m], n] &
But this is a very crude way of solving this problem.
Do you have any other ideas?