Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSIn particular, we note that L 0 n(x) =L n (x). As discussed in the previous section,L n (x) is a polynomial of order n and so it follows that L m n (x) is also. The first fewassociated Laguerre polynomials are easily found using (18.119):L m 0 (x) =1,L m 1 (x) =−x + m +1,2!L m 2 (x) =x 2 − 2(m +2)x +(m +1)(m +2),3!L m 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, thatL m n (x) =n∑(−1) k (n + m)!k!(n − k)!(k + m)! xk . (18.120)k=018.8.1 Properties of associated Laguerre polynomialsThe properties of the associated Laguerre polynomials follow directly from thoseof the ordinary Laguerre polynomials through the definition (18.119). We shalltherefore only briefly outline the most useful results here.Rodrigues’ formulaA Rodrigues’ formula for the associated Laguerre polynomials is given byL m n (x) = ex x −m d nn! dx n (xn+m e −x ). (18.121)It can be proved by evaluating the nth derivative using Leibnitz’ theorem (seeexercise 18.7).Mutual orthogonalityIn section 17.4, we noted that the associated Laguerre equation could be transformedinto a Sturm–Liouville one with p = x m+1 e −x , q =0,λ = n and ρ = x m e −x ,and its natural interval is thus [0, ∞]. Since the associated Laguerre polynomialsL m n (x) are solutions of the equation and are regular at the end-points, thosewith the same m but differing values of the eigenvalue λ = n must be mutuallyorthogonal over this interval with respect to the weight function ρ = x m e −x ,i.e.∫ ∞0L m n (x)L m k (x)x m e −x dx =0 ifn ≠ k.This result may also be proved directly using the Rodrigues’ formula (18.121), asmay the normalisation condition when k = n.622