The Associated Laguerre Polynomials and Functions

The sth derivative of the rth Laguerre polynomial is called the associated Laguerre polynomial of degree r — s and order s  

The differential equation satisfied by L (p) is found by differ-sntiating Equation 20-4 to be 

On comparing this with Equation 18-37 obtained in the treatment of the r equation for the hydrogen atom by the polynomial method, we see that the two equations are identical when kn+fKp) is identified with L(p) and the parameter X is replaced by its characteristic value n. The polynomials obtained in the solution of the r equation for the hydrogen atom are hence the associated Laguerre polynomials of degree n — l — 1 and of order 21 + 1. Moreover, the wave functions in r are, except for normalizing factors, the functions 

These functions are called the associated Laguerre functions. We shall discuss them in detail in succeeding sections. 

It is easily shown from Equation 20-1 that the generating function for the associated Laguerre polynomials of order s is1 

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Problem 20-2. Derive relations for the associated Laguerre polynomials and functions corresponding to those of Equations 20-2 and 20-3. 

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