RECURRENCE RELATIONS AND SPECIAL PROPERTIES
The Legendre polynomial generating function provides a convenient way of deriving the recurrence relations and some special properties. If our generating function (Eq. (12.4)) is differentiated with respect to t , we obtain
By substituting Eq. (12.4) into this and rearranging terms, we have
The left-hand side is a power series in t . Since this power series vanishes for all values of t , the coefﬁcient of each power of t is equal to zero; that is, our power series is unique (Section 5.7). These coefﬁcients are found by separating the individual summations and using distinctive summation indices:
Now, letting m = n + 1,s = n − 1, we ﬁnd
.This is another three-term recurrence relation, similar to (but not identical with) the recurrence relation for Bessel functions. With this recurrence relation we may easily construct the higher Legendre polynomials. If we take n = 1 and insert the easily found values of P0 (x ) and P1 (x ) , we obtain
More information about the behavior of the Legendre polynomials can be obtained if we now differentiate Eq. (12.4) with respect to x . This gives
As before, the coefﬁcient of each power of t is set equal to zero and we obtain
A more useful relation may be found by differentiating Eq. (12.17) with respect to x and multiplying by 2. To this we add (2n + 1) times Eq. (12.22), canceling the Pn′ term. The result is
From Eqs. (12.22) and (12.23) numerous additional equations may be developed,5 including
The previous equations, Eqs. (12.22) to (12.27), are all ﬁrst-order ODEs, but with polynomials of two different indices. The price for having all indices alike is a second-order differential equation. Equation (12.28) is Legendre’s ODE. We now see that the polynomials Pn (x ) generated by the power series for (1 − 2xt + t 2 )−1/2 satisfy Legendre’s equation, which, of course, is why they are called Legendre polynomials.
In Eq. (12.28) differentiation is with respect to x(x = cos θ). Frequently, we encounter Legendre’s equation expressed in terms of differentiation with respect to θ :
Our generating function provides still more information about the Legendre polynomials.
If we set x = 1, Eq. (12.4) becomes
using a binomial expansion or the geometric series, Example 5.1.1. But Eq. (12.4) for x = 1 deﬁnes
Comparing the two series expansions (uniqueness of power series, Section 5.7), we have
Pn (1) = 1. (12.31)
If we let x =−1 in Eq. (12.4) and use
this shows that
For obtaining these results, we ﬁnd that the generating function is more convenient than the explicit series form, Eq. (12.8).
If we take x = 0 in Eq. (12.4), using the binomial expansion
These results also follow from Eq. (12.8) by inspection.
Some of these results are special cases of the parity property of the Legendre polynomials.
We refer once more to Eqs. (12.4) and (12.8). If we replace x by −x and t by −t ,the generating function is unchanged. Hence
Comparing these two series, we have
that is, the polynomial functions are odd or even (with respect to x = 0,θ = π/2) according to whether the index n is odd or even. This is the parity,7 or reﬂection, property that plays such an important role in quantum mechanics. For central forces the index n is a measure of the orbital angular momentum, thus linking parity and orbital angular momentum.
This parity property is conﬁrmed by the series solution and for the special values tabulated in Table 12.1. It might also be noted that Eq. (12.37) may be predicted by inspection of Eq. (12.17), the recurrence relation. Speciﬁcally, if Pn−1 (x ) and xPn (x ) are even, then Pn+1 (x ) must be even.
Upper and Lower Bounds for Pn(cos θ)
Finally, in addition to these results, our generating function enables us to set an upper limit on |Pn (cos θ)|.Wehave
This series, Eq. (12.39b), is clearly a maximum when θ = 0 and cos mθ = 1. But for x = cos θ = 1, Eq. (12.31) shows that Pn (1) = 1. Therefore
A fringe beneﬁt of Eq. (12.39b) is that it shows that our Legendre polynomial is a linear combination of cos mθ . This means that the Legendre polynomials form a complete set for any functions that may be expanded by a Fourier cosine series (Section 14.1) over the interval [0,π ].
Legendre’s ODE (12.28) may be written in the form
showing that Pn (x ) and Pm (x ) are orthogonal for the interval [−1, 1]. This orthogonality may also be demonstrated by using Rodrigues’ deﬁnition of Pn (x ).We shall need to evaluate the integral (Eq. (12.41)) when n = m. Certainly it is no longer zero. From our generating function,
Integrating from x =−1to x =+1, we have
the cross terms in the series vanish by means of Eq. (12.42). Using y = 1 − 2tx + t 2 , dy =−2td x , we obtain
Expanding this in a power series gives us
Comparing power-series coefﬁcients of Eqs. (12.44) and (12.46), we must have
Combining Eq. (12.42) with Eq. (12.47) we have the orthonormality condition
We shall return to this result, when we construct the orthonormal spherical harmonics.
Expansion of Functions, Legendre Series
In addition to orthogonality, the Sturm–Liouville theory implies that the Legendre polynomials form a complete set. Let us assume, then, that the series
converges in the mean (Section 10.4) in the interval [−1, 1]. This demands that f(x ) and f ′ (x ) be at least sectionally continuous in this interval. The coefﬁcients an are found by multiplying the series by Pm (x ) and integrating term by term. Using the orthogonality property expressed in Eqs. (12.42) and (12.48), we obtain
We replace the variable of integration x by t and the index m by n. Then, substituting into Eq. (12.49), we have
This expansion in a series of Legendre polynomials is usually referred to as a Legendre series. Its properties are quite similar to the more familiar Fourier series . In particular, we can use the orthogonality property (Eq. (12.48)) to show that the series is unique.
On a more abstract (and more powerful) level, Eq. (12.51) gives the representation of f(x ) in the vector space of Legendre polynomials .
From the viewpoint of integral transforms, Eq. (12.50) may be considered a ﬁnite Legendre transform of f(x ). Equation (12.51) is then the inverse transform. It may also be interpreted in terms of the projection operators of quantum theory. We may take Pm in
as an (integral) operator, ready to operate on f(t ).(The f(t ) would go in the square bracket as a factor in the integrand.) Then, from Eq. (12.50),
The operator Pm projects out the mth component of the function f .
Equation (12.3), which leads directly to the generating function deﬁnition of Legendre polynomials, is a Legendre expansion of 1/r1 . This Legendre expansion of 1/r1 or 1/r12 . Going beyond a Coulomb ﬁeld, the 1 /r12 is often replaced by a potential V(|r1 − r2 |), and the solution of the problem is again effected by a Legendre expansion.
The Legendre series, Eq. (12.49), has been treated as a known function f(x ) that we arbitrarily chose to expand in a series of Legendre polynomials. Sometimes the origin and nature of the Legendre series are different. In the next examples we consider unknown functions we know can be represented by a Legendre series because of the differential equation the unknown functions satisfy. As before, the problem is to determine the unknown coefﬁcients in the series expansion. Here, however, the coefﬁcients are not found by Eq. (12.50). Rather, they are determined by demanding that the Legendre series match a known solution at a boundary. These are boundary value problems.