I tried to define a simple rule defining how λ acts on `ψ[n]`:

    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 `n` times:

    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?