SPECIAL FUNCTIONS
In particular, we note thatL^0 n(x)=Ln(x). As discussed in the previous section,Ln(x) is a polynomial of ordernand so it follows thatLmn(x) is also. The first few
associated Laguerre polynomials are easily found using (18.119):
Lm 0 (x)=1,Lm 1 (x)=−x+m+1,2!Lm 2 (x)=x^2 −2(m+2)x+(m+1)(m+2),3!Lm 3 (x)=−x^3 +3(m+3)x^2 −3(m+2)(m+3)x+(m+1)(m+2)(m+3).Indeed, in the general case, one may show straightforwardly, from the definition
(18.119) and the expression (18.111) for the ordinary Laguerre polynomials, that
Lmn(x)=∑nk=0(−1)k(n+m)!
k!(n−k)!(k+m)!xk. (18.120)18.8.1 Properties of associated Laguerre polynomialsThe properties of the associated Laguerre polynomials follow directly from those
of the ordinary Laguerre polynomials through the definition (18.119). We shall
therefore only briefly outline the most useful results here.
Rodrigues’ formulaA Rodrigues’ formula for the associated Laguerre polynomials is given by
Lmn(x)=exx−m
n!dn
dxn(xn+me−x). (18.121)It can be proved by evaluating thenth derivative using Leibnitz’ theorem (see
exercise 18.7).
Mutual orthogonalityIn section 17.4, we noted that the associated Laguerre equation could be trans-
formed into a Sturm–Liouville one withp=xm+1e−x,q=0,λ=nandρ=xme−x,
and its natural interval is thus [0,∞]. Since the associated Laguerre polynomials
Lmn(x) are solutions of the equation and are regular at the end-points, those
with the samembut differing values of the eigenvalueλ=nmust be mutually
orthogonal over this interval with respect to the weight functionρ=xme−x,i.e.
∫∞0Lmn(x)Lmk(x)xme−xdx=0 ifn=k.This result may also be proved directly using the Rodrigues’ formula (18.121), as
may the normalisation condition whenk=n.