Evaluating the closed form coefficients of Laguerre polynomials
For some numerical applications it can be useful to work with the closed form expression for Laguerre polynomials. The closed form is given by
| $$L_n(x) = \sum_{k=0}^n {n \choose k} \frac{(-1)^k}{k!} x^k$$ | (1) |
We can evaluate the coefficients with the help of the factorial formula
| $$\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$$ | (2) |
Since $n$ and $k$ are both integers, the coefficients are rational numbers. For most computational applications we will likely only need to work with floating point approximations of the coefficients.
When evaluating the coefficients using the factorial formula we should be careful to avoid overflow. For example if we try to compute $$n!$$ using default Fortran integers (32-bit), it will only be possible to calculate coefficients for $n \leq 12$.
To avoid such issues we can calculate the coefficients using the the gamma function $\Gamma$ and the logarithm thereof. This follows from the helpful observation that for integer arguments
| $$\Gamma(n+1) = n!$$ | (3) |
For the coefficients of $L_n(x)$, a straightforward Fortran implementation might look as follows:
pure function lagcoefs(n) result(a)
integer, intent(in) :: n
real :: a(0:n)
integer :: k
do k = 0, n
a(k) = (-1)**k * exp( &
log_gamma(real(n+1)) &
- 2*log_gamma(real(k+1)) &
- log_gamma(real(n-k+1)))
end do
end function
where we have used the properties of logarithms to transform the quotient into
subtraction. The input n should be a non-negative integer value.
The coefficients returned by the procedure are given in ascending order of
powers of $x$.
A similar procedure can be written for the derivative of the Laguerre polynomial. These are given be the expression
| $$L'_n(x) = \sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} k x^{k-1}$$ | (4) |
Here we should emphasize that the $i$-th derivative of a Laguerre polynomial is polynomial of order ${n - i}$ (a polynomial of order $n$ is described by $n+1$ coefficients).
The implementation can also be adapted easily to return the coefficients of the generalized Laguerre polynomials with the following closed form expression
| $$L_n^{(\alpha)}(x) = \sum_{k=0}^n (-1)^k \binom{n + \alpha}{ n - k} \frac{x^k}{k!}$$ | (5) |