Recurrence relation for harmonic oscillator wave functions

[squareroot]

1. Homework Statement
I've been using a recurrence relation from "Adv. in Physics"1966 Nr.57 Vol 15 . The relation is :

where R_nl are radial harmonic oscillator wave functions of form:

The problem is that I can't prove the relation above with the form of R_nl given by the author(above). I've been trying to check this using Mathematica. The form in which I need to use this is the following:

Here, I use the R_nl's as radial wave-functions for neutrons and protons and multiply the first relation with R^proton_nl and integrate over spherical coordinates.(P.S the R's where noted by g's ).

My questions is why would a relation like this only hold if the R's(or g, respectively) are normalized? Should't this relation hold for any radial wavefunctions? Where does the normalization condition comes from?

The above relations are equivalent with let's say

$$ | \alpha \rangle = a |\beta \rangle + b|\gamma \rangle $$

and by multiplying with the bra $| \alpha \rangle$ to the left one gets

$$ \langle \alpha | \alpha \rangle = a \langle \alpha | \beta \rangle + b \langle \alpha | \gamma \rangle $$

This is kind of my case above, but from my reasoning it should hold with no dependence on the normalization of the functions.

Thank you!

Homework Equations

The Attempt at a Solution

 

[DrClaude]

The associated Laguerre polynomials are not orthogonal for different values of $l$. It is the angular part that ensures the orthogonality of the wave functions for different $l$s.

 

[squareroot]

I know that. My question was related to the fact that I can check the recurrence relation ONLY if I normalize the R_nl's first. If I try to use the relation with non-normalized wave-functions the relation doesn't hold and I can't understand why the relation doesn't hold. That recurrence relation, from my reasoning, should hold even for non-normalized functions.