Implementing Generalized Laguerre Polynomials in Fortran

[d4n1el]

Hi!
Im trying to do some rather easy QM-calculations in Fortran.
To do that i need a routine that calculates the generalized Laguerre polynomials.

I just did the simplest implementation of the equation:
$$L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!}$$

I implemented this in the following way:

      SUBROUTINE LAGUERRE(n,l,r,u,arraylength)
      Implicit none
      INTEGER, INTENT(IN) :: arraylength,n
      real(kind=kind(0.d0)), INTENT(IN) :: l
      REAL(kind=kind(0.d0)), INTENT(IN),DIMENSION(arraylength) :: r
      REAL(kind=kind(0.d0)), INTENT(OUT),DIMENSION(arraylength) :: u
      REAL(kind=kind(0.d0)), DIMENSION(arraylength) :: temp
      Integer :: m
   
      temp=0.d0
      do m=0,n
         temp=temp+gamma(real(n)+l+1.0)*(-r)**m/
     +        (gamma(real(n)-real(m)+1.0)*gamma(l+real(m)+1.0)
     +        *gamma(real(m)+1.0))
      end do
      u=temp
      END SUBROUTINE LAGUERRE

The code is running and i do get results, but when i compare them with the results given in for example Mathematica, they seems quite strange.
So my questions are: Do you see some obvious mistakes? Do you think there are possibilities for numerical errors? (In such a case, do you have any advices on improvments?)

When i have been googling around, it looks like a lot of the numerical implementations are using recursive relations. What are the pro's and con's for doing this kind of calculation in such a manner?

 

[SteamKing]

What is gamma? Is it an array, a function, a subroutine?

 

[uart]

What is gamma? Is it an array, a function, a subroutine?

It looks like he's using it for factorial so it must the special function gamma (which reduces to factorial for integer values of the argument : gamma(n+1) = n! ).I just did the simplest implementation of the equation:
$$L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!}$$

I implemented this in the following way:

      SUBROUTINE LAGUERRE(n,l,r,u,arraylength)
      Implicit none
      INTEGER, INTENT(IN) :: arraylength,n
      real(kind=kind(0.d0)), INTENT(IN) :: l
      REAL(kind=kind(0.d0)), INTENT(IN),DIMENSION(arraylength) :: r
      REAL(kind=kind(0.d0)), INTENT(OUT),DIMENSION(arraylength) :: u
      REAL(kind=kind(0.d0)), DIMENSION(arraylength) :: temp
      Integer :: m
   
      temp=0.d0
      do m=0,n
         temp=temp+gamma(real(n)+l+1.0)*(-r)**m/
     +        (gamma(real(n)-real(m)+1.0)*gamma(l+real(m)+1.0)
     +        *gamma(real(m)+1.0))
      end do
      u=temp
      END SUBROUTINE LAGUERRE

Do you see some obvious mistakes?

Why are "r" and "u" declared as arrays yet you don't index them like arrays?

Where did the (l+k)! term come from?
Why isn't "r" squared.