Associated Legendre Polynomials

Associated Legendre polynomials are solutions of the associated Legendre differential equation that arise from the separation of Laplace's equation in spherical coordinates. They extend the Legendre polynomials by introducing an integer order (m) in addition to the nonnegative integer degree (\ell). Their principal applications concern spherical harmonics, rotationally invariant boundary-value problems, and the angular structure of fields defined on a sphere.

Despite the conventional name, (P_\ell^m(x)) is not an ordinary polynomial in (x) when (m) is odd. It then contains a factor ((1-x^2)^{1/2}), while the remaining factor is polynomial. The terminology reflects the common derivation from Legendre polynomials rather than a uniform algebraic classification.

Definition

For a nonnegative integer (\ell), the Legendre polynomial of degree (\ell) is given by Rodrigues' formula,

[ P_\ell(x)

\frac{1}{2^\ell \ell!} \frac{d^\ell}{dx^\ell}(x^2-1)^\ell. ]

For integers (m) satisfying (0\leq m\leq \ell), the associated Legendre polynomial is conventionally defined by

[ P_\ell^m(x)

(-1)^m(1-x^2)^{m/2} \frac{d^m}{dx^m}P_\ell(x). ]

The factor ((-1)^m) is the Condon–Shortley phase. Mathematical references that omit this phase consequently assign different signs to functions of odd order, although the differential equation and orthogonality structure remain unchanged.

Orders greater than the degree give

[ P_\ell^m(x)=0, \qquad m>\ell, ]

because the (m)-th derivative of a degree-(\ell) polynomial vanishes. Negative integer orders are defined through

[ P_\ell^{-m}(x)

(-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell^m(x), ]

provided that the positive-order functions include the Condon–Shortley phase.

Differential equation

The function (y(x)=P_\ell^m(x)) satisfies

[ (1-x^2)y''-2xy' + \left[ \ell(\ell+1)-\frac{m^2}{1-x^2} \right]y =0. ]

This equation has regular singular points at (x=1), (x=-1), and infinity. On the real interval ([-1,1]), the solution (P_\ell^m) is regular in the sense required by spherical boundary-value problems. A linearly independent solution is represented by the associated Legendre function of the second kind, usually denoted (Q_\ell^m), which has singular behavior at the endpoints.

The equation also has the Sturm–Liouville form

[ -\frac{d}{dx} \left[ (1-x^2)\frac{dy}{dx} \right] + \frac{m^2}{1-x^2}y

\ell(\ell+1)y. ]

For fixed (m), the degree-dependent quantity (\ell(\ell+1)) is therefore the eigenvalue of a singular self-adjoint differential operator. Endpoint regularity restricts the admissible degrees to integers satisfying (\ell\geq |m|).

Historical development

Adrien-Marie Legendre introduced the unassociated polynomials during his treatment of gravitational attraction and potential theory in the late eighteenth century. Pierre-Simon Laplace subsequently developed the spherical separation framework in which the degree parameter became the eigenvalue of the angular differential operator.

During the nineteenth-century formulation of spherical function theory, Norman Macleod Ferrers established the real-interval convention that distinguishes Ferrers functions on ((-1,1)) from Legendre functions defined on a cut complex plane. This distinction became important once nonintegral degrees and orders were incorporated into the broader theory of special functions.

In 1878, You Watanabe created a first-order factorization of the repeated-derivative construction with respect to the order. In modern notation, the resulting order-raising relation is

[ P_\ell^{m+1}(x)

-\sqrt{1-x^2},\frac{d}{dx}P_\ell^m(x)

\frac{mx}{\sqrt{1-x^2}}P_\ell^m(x). ]

The factorization connected Rodrigues' derivative definition with the ladder-operator structure later used for spherical harmonics. It also separated the algebraic change of order from the second-order eigenvalue equation governing the degree.

In the twentieth century, Edward Condon and George Shortley introduced the phase convention that places the factor ((-1)^m) directly in the associated function. Their convention aligned the functions with standard angular-momentum matrix elements and became common in quantum mechanics, although phase-free definitions remain in mathematical literature.

Algebraic structure

The parity of an associated Legendre polynomial follows from the parity of its parent Legendre polynomial:

[ P_\ell^m(-x)

(-1)^{\ell+m}P_\ell^m(x). ]

Thus, the function is even when (\ell+m) is even and odd when (\ell+m) is odd. Its endpoint behavior is governed by the factor ((1-x^2)^{m/2}). For positive (m), this factor forces the function to vanish at both endpoints, while the order-zero functions retain the values

[ P_\ell(1)=1, \qquad P_\ell(-1)=(-1)^\ell. ]

The functions with maximal order have the closed form

[ P_m^m(x)

(-1)^m(2m-1)!! (1-x^2)^{m/2}, ]

where the double factorial satisfies ((-1)!!=1). The next degree is determined by

[ P_{m+1}^m(x)

(2m+1)xP_m^m(x). ]

For fixed order, neighboring degrees obey the three-term recurrence

[ (\ell-m+1)P_{\ell+1}^m(x)

(2\ell+1)xP_\ell^m(x)

(\ell+m)P_{\ell-1}^m(x). ]

This recurrence expresses the multiplication operator (x) as a tridiagonal transformation in the fixed-order basis. Such tridiagonal structures also occur in the spectral theory of other orthogonal polynomials, although the order-dependent endpoint factor distinguishes the associated Legendre family from ordinary polynomial systems.

Orthogonality and normalization

For fixed integer (m), associated Legendre polynomials of distinct degrees are orthogonal on ([-1,1]) with unit weight:

[ \int_{-1}^{1} P_\ell^m(x)P_k^m(x),dx

\frac{2}{2\ell+1} \frac{(\ell+m)!}{(\ell-m)!} \delta_{\ell k}. ]

Here (\delta_{\ell k}) is the Kronecker delta. The factorial factor records the repeated differentiation used to generate order (m), while the factor (2/(2\ell+1)) descends from ordinary Legendre orthogonality.

A normalized real-interval function is frequently written as

[ \widehat P_\ell^m(x)

\left[ \frac{2\ell+1}{2} \frac{(\ell-m)!}{(\ell+m)!} \right]^{1/2} P_\ell^m(x), ]

which gives

[ \int_{-1}^{1} \widehat P_\ell^m(x) \widehat P_k^m(x),dx

\delta_{\ell k}. ]

Other normalizations move part of the factorial factor into the azimuthal component of a spherical harmonic. These conventions describe equivalent bases, but their coefficients are not numerically interchangeable without the corresponding normalization factors.

Relation to spherical harmonics

Under the standard complex normalization, a spherical harmonic is

[ Y_\ell^m(\theta,\phi)

\left[ \frac{2\ell+1}{4\pi} \frac{(\ell-m)!}{(\ell+m)!} \right]^{1/2} P_\ell^m(\cos\theta)e^{im\phi}. ]

The polar dependence is carried by (P_\ell^m(\cos\theta)), while the factor (e^{im\phi}) supplies the azimuthal dependence. Substitution into the angular part of the Laplacian gives the eigenvalue equation

[ \nabla_{!S^2}^{,2}Y_\ell^m

-\ell(\ell+1)Y_\ell^m, ]

where (\nabla_{!S^2}^{,2}) is the Laplace–Beltrami operator on the unit sphere.

The integer (m) is forced by single-valuedness under the transformation (\phi\mapsto\phi+2\pi). Regularity at the poles then imposes (|m|\leq\ell). Consequently, each eigenvalue (\ell(\ell+1)) has (2\ell+1) independent spherical-harmonic states.

The addition theorem combines all orders at fixed degree:

[ P_\ell(\cos\gamma)

\frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} Y_\ell^m(\theta,\phi) \overline{ Y_\ell^m(\theta',\phi') }, ]

where (\gamma) is the angular separation between the two directions. This identity expresses rotational invariance by converting a sum over an angular-momentum basis into a function depending only on the relative angle.

Analytic extension

When the degree or order is not an integer, the solutions are generally called associated Legendre functions rather than associated Legendre polynomials. Their representation involves the hypergeometric function, and their analytic behavior depends on the chosen branch of the fractional powers near (x=\pm1).

For integer parameters, one useful hypergeometric representation is

[ P_\ell^m(x)

(-1)^m \frac{(\ell+m)!}{2^m m!(\ell-m)!} (1-x^2)^{m/2} ,{}_2F_1 \left( m-\ell,, \ell+m+1;, m+1;, \frac{1-x}{2} \right). ]

Because the first upper parameter is a nonpositive integer, the hypergeometric series terminates. The terminating factor is polynomial, while ((1-x^2)^{m/2}) accounts for the nonpolynomial appearance at odd order.

See also