The Laguerre Polynomials

Using this result, we obtain the constant necessary to normalize the part of the wave function which depends on d. The final form for 0( ) is 

Problem 19-1. Prove that the definition of the Legendre polynomials 

Problem 19-2. Derive the following relations involving the associated 

The Laguerre Polynomials.—The Laguerre polynomials of a variable p, within the limits 0 p may be defined by means of the generating function 

To find the differential equation satisfied by these polynomials Lr(p), we follow the now familiar procedure of differentiating the 

The associated Laguerre polynomials are related to the Laguerre polynomials L (x) by the expression 

The Laguerre polynomials L (x) are normalized to unity see (6.5.2). We note that, for positive integers a, the associated Laguerre polynomials may be obtained from the Laguerre polynomials 

It is important to realize that different conventions are used for the associated Laguerre polynomials. Thus, the associated Laguerre polynomials are frequently defined as 

The following special cases for integers and half integers are important  

Here the factorial function is defined for zero and all positive integers 

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As the isotropic Raman and CARS spectra may be expressed via ao(co) by virtue of the Laguerre polynomials orthogonality, we have... 

Undetermined coefficients 4 can be easily found using the initial condition (6.3) and orthogonality of the Laguerre polynomials [37]... 

Combining equations (F.3) and (F.4), we obtain the formula for the Laguerre polynomials... 

Since the Laguerre polynomial Lk p) divided by k is the coefficient of 5 in the expansion (F.l) of the generating function, we have... 

The associated Laguerre polynomials L p) are defined in terms of the Laguerre polynomials by... 

The Laguerre Polynomials. Tiic Lagucrre polynomials Lh x) are defined for a positive integer and x a positive real number by the equation... 

The wavefunctions are given in terms of associated Laguerre polynomials, and using the approximate form of the Laguerre polynomials for large arguments, we can write an approximate, unnormalized wavefunction as1... 

Working out the parabolic wavefunctions in terms of the Laguerre polynomials is useful in the analytic calculation of the Stark effect using perturbation theory. However, it is not useful in very strong electric fields. Here we outline a more general procedure which is valid in strong fields and lends itself to numerical computations. If we replace jq( ) and u2(rj) in Eqs. (6.8a) and (6.8b) by1-3 15... 

The Laguerre polynomials Ln(r) become orthonormal polynomials only after they are multiplied by a weighting function ( ) 1 exp( r) [11] ... 

The Laguerre polynomials form part of the solution of the R equation in the quantum mechanical treatment of the hydrogen atom. One of these polynomials is defined by... 

At this point, it will be useful to recall some of the properties of the Laguerre polynomials. The Laguerre polynomials are a set of orthogonal polynomials that satisfy the differential equation (Gradshteyn and Ryzhik, 1980, pp. 1037-1039)... 

While the radial function in Equation (38) with integer nr involves the Laguerre polynomials, the corresponding function in Equation (92) with non-integer vr becomes an infinite series. This poses no difficulty in the accurate evaluation of its zeros as already discussed in Section 2.1, in connection with the s-states. The reader may also take notice of the f-dependence of the function and the eigenenergies in Equations (92) and... 

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