replacement - Efficient way of setting up a rule

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?

Answer

If you look at the full form versions of λ ψ[n] and λ^k ψ[n]

λ ψ[n] // FullForm

Times[λ, ψ[n]]

λ^k ψ[n] // FullForm

Times[Power[λ, k],ψ[n]]

you see that the second expression can't match your rule because it contains Power. Therefore, as Daniel Lichtblau suggests, you need to have two rules.

rules = {λ ψ[n_] -> α[1 + n] ψ[1 + n],

         λ^j_ ψ[n_] -> Product[α[s], {s, n + 1, n + j}] ψ[j + n]};

Now you can get both

λ ψ[n] /. rules

α[1 + n] ψ[1 + n]

and

λ^k ψ[n] /. rules

Product[α[s], {s, n + 1, n + k}] ψ[k + n]