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ASSOCIATED LEGENDRE FUNCTIONS

When Helmholtz’s equation is separated in spherical polar coordinates, one of the separated ODEs is the associated Legendre equation

With x = cos θ , this becomes

If the azimuthal separation constant m² = 0, we have Legendre’s equation, Eq. (12.28).
The regular solutions   (with m not necessarily zero, but an integer) are

          (12.73a)

with m ≥ 0 an integer

One way of developing the solution of the associated Legendre equation is to start with the regular Legendre equation and convert it into the associated Legendre equation by using multiple differentiation. These multiple differentiations are suggested by Eq. (12.73a). 
We take Legendre’s equation

(1 − x ²)u′′ − 2x(m + 1)u′ + (n − m)(n + m + 1)u = 0,       (12.75)

where

(12.76)

Equation (12.74) is not self-adjoint. To put it into self-adjoint form and convert the weighting function to 1, we replace u(x ) by

      (12.73b)

Solving for u and differentiating, we obtain

Substituting into Eq. (12.74), we find that the new function v satisfies the self-adjoint ODE

which is the associated Legendre equation; it reduces to Legendre’s equation when m is set equal to zero. Expressed in spherical polar coordinates, the associated Legendre equation is

Associated Legendre Polynomials 

The regular solutions, relabeled

       (12.73c)

These are the associated Legendre functions. Since the highest power of x in P_n (x ) is x ⁿ ,wemusthave m ≤ n (or the m-fold differentiation will drive our function to zero).
In quantum mechanics the requirement that m ≤ n has the physical interpretation that the expectation value of the square of the z component of the angular momentum is less than or equal to the expectation value of the square of the angular momentum vector L,

From our definition of the associated Legendre functions 

           (12.82)

A generating function for the associated Legendre functions is obtained, via Eq. (12.71), from that of the ordinary Legendre polynomials:

        (12.83)

Comparing coefficients of powers of t in these power series, we obtain the recurrence relation

so upon inserting Eq. (12.84) we get the recursion

Table 12.2 Associated Legendre Functions

Example 12.5.1 LOWEST ASSOCIATED LEGENDRE POLYNOMIALS

Example 12.5.2 SPECIAL VALUES

For x = 1we use

in Eq. (12.84) and find

(12.90)

where

for s = 0 and

Recurrence Relations

As expected and already seen, the associated Legendre functions satisfy recurrence relations. Because of the existence of two indices instead of just one, we have a wide variety of recurrence relations:

These relations, and many other similar ones, may be verified by use of the generating function (Eq. (12.4)), by substitution of the series solution of the associated Legendre equation (12.79) or reduction to the Legendre polynomial recurrence relations, using Eq. (12.73c). As an example of the last method, consider Eq. (12.93). It is similar to Eq. (12.23):

         (12.95)

Let us differentiate this Legendre polynomial recurrence relation m times to obtain

Now multiplying by (1 − x ² )^(m+1)/2 and using the definition of P_n (x ), we obtain the first part of Eq. (12.93).

Parity

The parity relation satisfied by the associated Legendre functions may be determined by examination of the defining equation (12.73c). As x →−x , we already know that P_n (x ) contributes a (−1)ⁿ .The m-fold differentiation yields a factor of (−1)^m . Hence we have

  (12.97)

A glance at Table 12.2 verifies this for 1 ≤ m ≤ n ≤ 4.
Also, from the definition in Eq. (12.73c),

        (12.98)

Orthogonality 

The orthogonality of the  follows from the ODE, just as for the P_n (x ) (Section 12.3), if m is the same for both functions. However, it is instructive to demonstrate the orthogonality by another method, a method that will also provide the normalization constant.
Using the definition in Eq. (12.73c) and Rodrigues’ formula (Eq. (12.65)) for P_n (x ),we find

The integrand on the right-hand side is now expanded by Leibniz’ formula to give

Since the term X^m contains no power of x greater than x^2m ,wemusthave

q + m − i ≤ 2m                    (12.102)

or the derivative will vanish. Similarly,

p + m + i ≤ 2p.               (12.103)

Adding both inequalities yields

q ≤ p,                                 (12.104)

which contradicts our assumption that p< q . Hence, there is no solution for i and the integral vanishes. The same result obviously will follow if p> q .
For the remaining case, p = q , we have the single term corresponding to i = q − m.
Putting Eq. (12.101) into Eq. (12.100), we have

Since

Eq. (12.105) reduces to

The integral on the right is just

(compare Exercise 8.4.9). Combining Eqs. (12.108) and (12.109), we have the orthogonality integral,

or, in spherical polar coordinates,

The orthogonality of the Legendre polynomials is a special case of this result, obtained by setting m equal to zero; that is, for m = 0, Eq. (12.110) reduces to Eqs. (12.47) and (12.48). In both Eqs. (12.110) and (12.111), our Sturm–Liouville theory of Chapter 10 could provide the Kronecker delta. A special calculation, such as the analysis here, is required for the normalization constant.
The orthogonality of the associated Legendre functions over the same interval and with the same weighting factor as the Legendre polynomials does not contradict the uniqueness of the Gram–Schmidt construction of the Legendre polynomials.
Table 12.2 suggests that dx may be written as

where we defined earlier

The functions  may be constructed by the Gram–Schmidt procedure with the weighting function w(x ) = (1 − x ² )^m .
It is possible to develop an orthogonality relation for associated Legendre functions of the same lower index but different upper index. We find

Note that a new weighting factor, (1 − x ² )⁻¹ , has been introduced. This relation is a mathematical curiosity. In physical problems with spherical symmetry solutions of Eqs. (12.80) and (9.64) appear in conjunction with those of Eq. (9.61), and orthogonality of the azimuthal dependence makes the two upper indices equal and always leads to Eq. (12.111).