13.2 Laguerre Functions 837 13.2 LAGUERRE FUNCTIONS Differential Equation — Laguerre Polynomials If we start with the appropriate generating function, it is possible to develop the Laguerre polynomials in analogy with the Hermite polynomials. Alternatively, a series solution may be developed by the methods of Section 9.5. Instead, to illustrate a different technique, let us start with Laguerre’s ODE and obtain a solution in the form of a contour integral, as we did with the integral representation for the modified Bessel function Kν(x) (Section 11.6). From this integral representation a generating function will be derived. Laguerre’s ODE (which derives from the radial ODE of Schrödinger’s PDE for the hy- drogen atom) is xy′′(x) (1 x)y′(x) ny(x) 0. (13.52) + − + = We shall attempt to represent y,orratheryn,sincey will depend on the parameter n, anonnegativeinteger,bythecontourintegral xz/(1 z) 1 e− − yn(x) dz (13.53a) = 2πi (1 z)zn 1 ! − + and demonstrate that it satisfies Laguerre’s ODE. The contour includes the origin but does not enclose the point z 1. By differentiating the exponential in Eq. (13.53a) we obtain = xz/(1 z) 1 e− − y′ (x) dz, (13.53b) n =−2πi (1 z)2zn ! − xz/(1 z) 1 e− − y′′(x) dz. (13.53c) n = 2πi (1 z)3zn 1 ! − − Substituting into the left-hand side of Eq. (13.52), we obtain 1 x 1 x n xz/(1 z) − e− − dz, 2πi (1 z)3zn 1 − (1 z)2zn + (1 z)zn 1 ! " − − − − + # which is equal to 1 d e xz/(1 z) − − dz. (13.54) −2πi dz (1 z)zn ! " − # If we integrate our perfect differential around a closed contour (Fig. 13.3), the integral will vanish, thus verifying that yn(x) (Eq. (13.53a)) is a solution of Laguerre’s equation. 5 It has become customary to define Ln(x),theLaguerrepolynomial(Fig.13.4),by xz/(1 z) 1 e− − Ln(x) dz. (13.55) = 2πi (1 z)zn 1 ! − + 5 Other definitions of Ln(x) are in use. The definitions here of the Laguerre polynomial Ln(x) and the associated Laguerre k polynomial Ln(x) agree with AMS-55, Chapter 22. (For the full ref. see footnote 4 in Chapter 5 or the General References at book’s end.) 838 Chapter 13 More Special Functions FIGURE 13.3 Laguerre polynomial contour. FIGURE 13.4 Laguerre polynomials. This is exactly what we would obtain from the series xz/(1 z) e− − ∞ n g(x,z) Ln(x)z , z < 1, (13.56) = 1 z = | | n 0 − $= n 1 if we multiplied g(x,z) by z− − and integrated around the origin. As in the development 1 of the calculus of residues (Section 7.1), only the z− term in the series survives. On this basis we identify g(x,z) as the generating function for the Laguerre polynomials. 13.2 Laguerre Functions 839 With the transformation xz s x s x, or z − , (13.57) 1 z = − = s − x n s e s e− Ln(x) ds, (13.58) = 2πi (s x)n 1 ! − + the new contour enclosing the point s x in the s-plane. By Cauchy’s integral formula = (for derivatives), x n e d n x Ln(x) x e− (integral n), (13.59) = n dxn ! % & giving Rodrigues’ formula for Laguerre polynomials. From these representations of Ln(x) we find the series form (for integral n), n 2 2 2 ( 1) n n n 1 n (n 1) n 2 n Ln(x) − x x − − x − ( 1) n = n − 1 + 2 −···+ − ! ! " ! ! # n n ( 1)mn xm ( 1)n sn xn s − ! − − ! − (13.60) = (n m) m m = (n s) (n s) s m 0 s 0 $= − ! ! ! $= − ! − ! ! and the specific polynomials listed in Table 13.2 (Exercise 13.2.1). Clearly, the defini- tion of Laguerre polynomials in Eqs. (13.55), (13.56), (13.59), and (13.60) are equivalent. Practical applications will decide which approach is used as one’s starting point. Equa- tion (13.59) is most convenient for generating Table 13.2, Eq. (13.56) for deriving recur- sion relations from which the ODE (13.52) is recovered. By differentiating the generating function in Eq. (13.56) with respect to x and z,we obtain recurrence relations for the Laguerre polynomials as follows. Using the product rule for differentiation we verify the identities ∂g ∂g (1 z)2 (1 x z)g(x, z), (z 1) zg(x, z). (13.61) − ∂z = − − − ∂x = Table 13.2 Laguerre Polynomials L (x) 1 0 = L (x) x 1 1 =− + 2 L (x) x2 4x 2 ! 2 = − + 3 L (x) x3 9x2 18x 6 ! 3 =− + − + 4 L (x) x4 16x3 72x2 96x 24 ! 4 = − + − + 5 L (x) x5 25x4 200x3 600x2 600x 120 ! 5 =− + − + − + 6 L (x) x6 36x5 450x4 2400x3 5400x2 4320x 720 ! 6 = − + − + − + 840 Chapter 13 More Special Functions Writing the left-hand and right-hand sides of the first identity in terms of Laguerre polyno- mials using Eq. (13.56) we obtain n (n 1)Ln 1(x) 2nLn(x) (n 1)Ln 1(x) z + − n + − + − $' ( n (1 x)Ln(x) Ln 1(x) z . − = n − − $' ( Equating coefficients of zn yields (n 1)Ln 1(x) (2n 1 x)Ln(x) nLn 1(x). (13.62) + + = + − − − To get the second recursion relation we use both identities of Eqs. (13.61) to verify the third identity, ∂g ∂g ∂(zg) x z z , (13.63) ∂x = ∂z − ∂z which, when written similarly in terms of Laguerre polynomials, is seen to be equivalent to xLn′ (x) nLn(x) nLn 1(x). (13.64) = − − Equation (13.61), modified to read 1 Ln 1(x) 2Ln(x) Ln 1(x) (1 x)Ln(x) Ln 1(x) , (13.65) + = − − − n 1 + − − + ' ( for reasons of economy and numerical stability, is used for computation of numerical val- ues of Ln(x).ThecomputerstartswithknownnumericalvaluesofL0(x) and L1(x),Ta- ble 13.2, and works up step by step. This is the same technique discussed for computing Legendre polynomials, Section 12.2. Also, from Eq. (13.56) we find 1 ∞ n ∞ n g(0,z) z Ln(0)z , = 1 z = = n 0 n 0 − $= $= which yields the special values of Laguerre polynomials Ln(0) 1. (13.66) = As is seen from the form of the generating function, from the form of Laguerre’s ODE, or from Table 13.2, the Laguerre polynomials have neither odd nor even symmetry under the parity transformation x x. →− The Laguerre ODE is not self-adjoint, and the Laguerre polynomials Ln(x) do not by themselves form an orthogonal set. However, following the method of Section 10.1, if we x multiply Eq. (13.52) by e− (Exercise 10.1.1) we obtain ∞ x e− Lm(x)Ln(x) dx δmn. (13.67) = )0 13.2 Laguerre Functions 841 This orthogonality is a consequence of the Sturm–Liouville theory, Section 10.1. The nor- malization follows from the generating function. It is sometimes convenient to define or- thogonalized Laguerre functions (with unit weighting function) by x/2 ϕn(x) e− Ln(x). (13.68) = Our new orthonormal function, ϕn(x),satisfiestheODE 1 x xϕ′′(x) ϕ′ (x) n ϕn(x) 0, (13.69) n + n + + 2 − 4 = * + which is seen to have the (self-adjoint) Sturm–Liouville form. Note that the interval (0 x< ) was used because Sturm–Liouville boundary conditions are satisfied at its ≤ ∞ endpoints. Associated Laguerre Polynomials In many applications, particularly in quantum mechanics, we need the associated Laguerre polynomials defined by6 k k k d Ln(x) ( 1) Ln k(x). (13.70) = − dxk + From the series form of Ln(x) we verify that the lowest associated Laguerre polynomials are given by Lk(x) 1, 0 = Lk(x) x k 1, 1 =− + + x2 (k 2)(k 1) Lk(x) (k 2)x + + . (13.71) 2 = 2 − + + 2 In general, n (n k) Lk (x) ( 1)m + ! xm,k> 1. (13.72) n = − (n m) (k m) m − m 0 $= − ! + ! ! AgeneratingfunctionmaybedevelopedbydifferentiatingtheLaguerregeneratingfunc- tion k times to yield k xz/(1 z) k k d e− − k ∞ d n k ∞ k n k ( 1) ( 1) Ln k(x)z + Ln(x)z + − dxk 1 z = − dxk + = n 0 n 0 − $= $= z k exz/(1 z) − . = 1 z 1 z * − + − 6 Lk k k k kLk Some authors use n k(x) (d /dx ) Ln k(x) .HenceourLn(x) ( 1) n k(x). + = [ + ] = − + 842 Chapter 13 More Special Functions From the last two members of this equation, canceling the common factor zk,weobtain e xz/(1 z) ∞ − − Lk (x)zn, z < 1. (13.73) (1 z)k 1 = n | | + n 0 − $= From this, for x 0, the binomial expansion = 1 ∞ k 1 ∞ − − ( z)n Lk (0)zn (1 z)k 1 = n − = n + n 0 * + n 0 − $= $= yields (n k) Lk (0) + !. (13.74) n = n k ! ! Recurrence relations can be derived from the generating function or by differentiating the Laguerre polynomial recurrence relations. Among the numerous possibilities are k k k (n 1)Ln 1(x) (2n k 1 x)Ln(x) (n k)Ln 1(x), (13.75) + + = + + − − + − k dLn(x) k k x nLn(x) (n k)Ln 1(x). (13.76) dx = − + − Thus, we obtain from differentiating Laguerre’s ODE once dLn′′ dLn′ dLn x L′′ L′ (1 x) n 0, dx + n − n + − dx + dx = and eventually from differentiating Laguerre’s ODE k times dk dk 1 dk 1 dk dk x L k − L k − L ( x) L n L . k n′′ k 1 n′′ k 1 n′ 1 k n′ k n 0 dx + dx − − dx − + − dx + dx = Adjusting the index n n k,wehavetheassociatedLaguerreODE → + d2Lk (x) dLk (x) x n (k 1 x) n nLk (x) 0.