Associated Legendre polynomials
Associated Legendre polynomials P_\ell^m(x) are a class of special functions in mathematics that generalize the ordinary Legendre polynomials P_\ell(x) by introducing an integer order parameter m, where \ell is the non-negative integer degree with |m| \leq \ell. They are defined explicitly as P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_\ell(x) for m \geq 0, and extended to negative orders via P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x).[1][2] These polynomials satisfy the associated Legendre differential equation (1 - x^2) y'' - 2x y' + \left[ \ell(\ell + 1) - \frac{m^2}{1 - x^2} \right] y = 0, which arises as a separation of variables in the Laplace equation under spherical coordinates.[1] For fixed m, the associated Legendre polynomials P_\ell^m(x) form an orthogonal basis over the interval [-1, 1] with respect to the weight function 1, satisfying \int_{-1}^{1} P_\ell^m(x) P_k^m(x) \, dx = \frac{2 (\ell + m)!}{ (2\ell + 1) (\ell - m)! } \delta_{\ell k}.[1] They exhibit various recurrence relations and generating functions that facilitate their computation and analysis.[2] In applications, associated Legendre polynomials are fundamental components of spherical harmonics Y_\ell^m(\theta, \phi), which provide complete orthogonal bases for functions on the sphere and solve the angular part of the Laplace equation in spherical coordinates.[1] In physics, they appear prominently in quantum mechanics as part of the hydrogen atom wavefunctions, describing the angular momentum states,[3] and in electromagnetism for multipole expansions of potentials.[4] Additionally, they find use in geophysics for modeling gravitational and magnetic fields,[5] as well as in fluid dynamics for problems involving spherical symmetry.[6]Definitions and Basic Forms
Standard Definition
The associated Legendre polynomials P_\ell^m(x), where \ell is the degree and m is the order, are defined for non-negative integers \ell and m satisfying m \leq \ell by the formula P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_\ell(x), with P_\ell(x) denoting the ordinary Legendre polynomial of degree \ell.[7] This expression arises in the context of separating variables in the Laplace equation in spherical coordinates, where x = \cos \theta and the factor (1 - x^2)^{m/2} accounts for the azimuthal dependence.[8] The parameter \ell represents the total angular momentum quantum number in quantum mechanics or the multipole order in potential theory, and it must be a non-negative integer with \ell \geq m. The order m is an integer ranging from 0 to \ell, capturing the projection along a specific axis; when m = 0, the associated Legendre polynomials reduce to the ordinary Legendre polynomials P_\ell^0(x) = P_\ell(x).[7] Adrien-Marie Legendre introduced the ordinary Legendre polynomials in 1782 as part of his work on the gravitational attraction of ellipsoids, laying the foundation for these functions.[9] The associated forms were developed shortly thereafter to address problems involving spherical symmetry and non-zero azimuthal orders, such as in the expansion of potentials around spheres.[9] This definition can be derived from the Rodrigues formula for the Legendre polynomials, P_\ell(x) = \frac{1}{2^\ell \ell!} \frac{d^\ell}{dx^\ell} (x^2 - 1)^\ell, by applying m additional differentiations to both sides and incorporating the (-1)^m (1 - x^2)^{m/2} prefactor, which follows from the structure of the associated Legendre differential equation and Leibniz's rule for higher derivatives.Closed-Form Expression
The closed-form expression for the associated Legendre polynomials P_\ell^m(x) is provided by the following explicit summation formula, which facilitates direct numerical evaluation for nonnegative integers \ell \geq m \geq 0: P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \sum_{k=0}^{\lfloor (\ell - m)/2 \rfloor} (-1)^k \binom{\ell}{k} \binom{2\ell - 2k}{\ell} \frac{(\ell - 2k)!}{2^\ell (\ell - m - 2k)!} x^{\ell - m - 2k}. This representation arises from the standard series expansion of the Legendre polynomial P_\ell(x) given in equation (18.5.8) of the NIST Handbook, P_\ell(x) = \sum_{k=0}^{\lfloor \ell/2 \rfloor} (-1)^k \frac{\binom{2\ell - 2k}{\ell} \binom{\ell}{k}}{2^\ell} x^{\ell - 2k}, combined with the defining relation in equation (14.3.7), P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_\ell(x), where the m-th derivative is applied term by term to the series for P_\ell(x), retaining only terms where the power of x exceeds or equals m and adjusting the coefficients accordingly.[10][7] The prefactor (1 - x^2)^{m/2} ensures the expression is suitable for |x| \leq 1, the domain of orthogonality, and introduces the necessary square-root branch for non-integer powers when m > 0, though the overall function remains a polynomial in x multiplied by this factor. The summation terminates after at most \lfloor (\ell - m)/2 \rfloor + 1 terms, making it computationally efficient for moderate \ell and m, as each term involves only binomial coefficients and factorials, which can be precomputed or evaluated recursively to avoid overflow in numerical implementations. For illustration, consider \ell = 2, m = 1. The upper limit is \lfloor (2-1)/2 \rfloor = 0, so the sum has a single term at k=0: \sum_{k=0}^{0} (-1)^0 \binom{2}{0} \binom{4}{2} \frac{2!}{2^2 (2-1-0)!} x^{2-1-0} = 1 \cdot 1 \cdot 6 \cdot \frac{2}{4 \cdot 1!} x = 3x. Thus, P_2^1(x) = (-1)^1 (1 - x^2)^{1/2} \cdot 3x = -3x \sqrt{1 - x^2}, which matches the known explicit form derived from the Rodrigues formula.[7]Alternative Notations
The associated Legendre polynomials, often denoted as P_\ell^m(x), exhibit variations in notation and conventions across mathematical, physical, and applied literature, primarily due to differences in phase factors and symbol placement. A prominent example is the Condon-Shortley phase, which introduces a factor of (-1)^m for m \geq 0 in the definition to ensure consistency with angular momentum operators in quantum mechanics.[11] This phase is included in standard references such as the Handbook of Mathematical Functions by Abramowitz and Stegun (equation 8.6.1), where the associated Legendre polynomial is defined with the leading (-1)^m.[12] In contrast, fields like geophysics and geodesy typically omit this phase in definitions of both the polynomials and related spherical harmonics, as it simplifies computations in potential theory without affecting orthogonality.[13] Symbolic notations also differ, with the most common form in physics and engineering being P_\ell^m(x), where \ell (the degree) is subscripted and m (the order) is superscripted.[1] Mathematical texts, such as the NIST Digital Library of Mathematical Functions (DLMF), reverse this to P^\mu_\nu(x), placing the order \mu as superscript and degree \nu as subscript, aligning with broader conventions for special functions.[7] Less frequently, the notation P_m^\ell(x) appears in some 19th- and early 20th-century works, interchanging the indices to emphasize the order first. Related spherical harmonics are denoted Y_\ell^m(\theta, \phi), incorporating the associated Legendre polynomials via Y_\ell^m \propto P_\ell^m(\cos \theta) e^{im\phi}, with the phase convention influencing the normalization. Ferrers functions represent another notational variant, specifically the associated Legendre functions of the first kind for -1 < x < 1, often denoted \mathsf{P}_\ell^m(x) to distinguish the real-valued form on the branch cut interval, as used in geophysical modeling for interior problems.[7] These are distinguished from the standard functions defined for |x| > 1 in some contexts. Historically, the notation for associated Legendre polynomials evolved from Adrien-Marie Legendre's 1782 introduction of ordinary Legendre polynomials P_\ell(x) (for m=0) in studies of gravitational potentials, initially without associated forms.[14] Pierre-Simon Laplace extended these to higher orders in the early 19th century for celestial mechanics, leading to the associated variants, though early texts like those by Carl Gustav Jacobi (1840s) used ad hoc symbols without standardized indices.[15] By the late 19th century, texts such as Whittaker and Watson's A Course of Modern Analysis (1902) adopted P_\nu^\mu(x), influencing 20th-century conventions toward the superscript-subscript dichotomy.| Source | Notation | Condon-Shortley Phase Included? | Notes |
|---|---|---|---|
| Abramowitz and Stegun (1964, Ch. 8) | P_l^m(x) | Yes, via (-1)^m factor | Standard in applied mathematics and physics; equation 8.6.1.[12] |
| NIST DLMF (2010, §14.3) | P^\mu_\nu(x) | Yes | Emphasizes general \nu, \mu; aligns with hypergeometric representations; distinguishes Ferrers functions for -1 < x < 1.[7] |
Extensions for Negative Parameters
Negative m Values
The associated Legendre polynomials for negative orders m, where - \ell \leq m < 0 and \ell is a non-negative integer, are defined in relation to those with positive orders via the formula P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x), where P_\ell^m(x) is the standard associated Legendre polynomial for positive m.[1] This relation ensures that the functions remain solutions to the associated Legendre differential equation and inherit key analytical properties from the positive-order case.[16] This extension arises primarily in the context of spherical harmonics, where the magnetic quantum number m ranges from -\ell to \ell to account for the full azimuthal dependence in spherical coordinates, as seen in quantum mechanics and electromagnetism.[16] The negative m values correspond to complex conjugates or phase-adjusted versions of positive m harmonics, enabling complete orthonormal bases on the sphere.[2] The relation can be derived from the general Rodrigues formula for associated Legendre polynomials, P_\ell^m(x) = \frac{(-1)^m}{2^\ell \ell!} (1 - x^2)^{m/2} \frac{d^{ \ell + m }}{dx^{ \ell + m }} (x^2 - 1)^\ell, by formally applying the formula for negative m and using integration by parts or properties of the differential operator to relate it back to the positive case, yielding the factorial prefactor.[16] Alternatively, the generating function approach, \left(1 - 2xt + t^2\right)^{-1/2} \exp\left( \frac{m}{2} \ln \frac{1 - t}{1 + t} \right) = \sum_{\ell = |m|}^\infty P_\ell^m(x) t^\ell, extends naturally to negative m through symmetry in the expansion coefficients.[1] Regarding normalization, the prefactor in the relation adjusts the L^2-norm such that the orthogonality integral for negative m takes the form \int_{-1}^1 P_\ell^{-m}(x) P_{\ell'}^{-m}(x) \, dx = \frac{2}{2\ell + 1} \frac{(\ell - |m|)!}{(\ell + |m|)!} \delta_{\ell \ell'}, which differs from the positive m case. This ensures consistency in the differential equation solutions, while applications like spherical harmonics expansions incorporate additional normalization factors to achieve uniform integrals over the sphere.[1][16]Negative ℓ Values
The associated Legendre polynomials can be extended to negative degrees \ell < 0 through specific identities that relate them to their positive-degree counterparts. In the standard convention for integer degrees, the function P_{-\ell-1}^m(x) for non-negative integer \ell and integer m with |m| \leq \ell is defined by the relation P_{-\ell-1}^m(x) = P_\ell^m(x), where P_\ell^m(x) is the associated Legendre polynomial of positive degree \ell. This equality holds because the parameter in the associated Legendre differential equation, \ell(\ell+1), remains unchanged when \ell is replaced by -\ell-1, leading to identical eigenvalue problems and thus equivalent solutions on the interval [-1, 1]. This derivation arises directly from the associated Legendre differential equation, (1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + \left[ \ell(\ell + 1) - \frac{m^2}{1 - x^2} \right] y = 0, which admits nonsingular solutions on [-1, 1] only for appropriate integer parameters, and the substitution \ell \to -\ell - 1 preserves the form of the equation and its bounded solutions. Alternative conventions may relate P_{-\ell}^m(x) to P_{\ell-1}^m(x), but the relation to -\ell-1 is the most widely adopted for ensuring consistency with hypergeometric representations and analytic properties.[7] In physical applications, such as the expansion of solutions to Laplace's equation in spherical coordinates or quantum mechanical angular momentum eigenfunctions, negative integer degrees \ell < 0 lack direct relevance, as the degree corresponds to non-negative integers representing physical quantities like orbital angular momentum.[17] However, these extensions hold mathematical utility in analytic continuations of the functions to complex or non-integer parameters, facilitating solutions to boundary value problems in more general geometries, such as toroidal domains or wedge-shaped regions.[18] For small negative degrees, the equivalence is evident in explicit computations. For \ell = 0 and m = 0, P_0^0(x) = 1, so P_{-1}^0(x) = 1. For \ell = 1 and m = 0, P_1^0(x) = x, yielding P_{-2}^0(x) = x. Similarly, for \ell = 1 and m = 1, P_1^1(x) = -(1 - x^2)^{1/2}, and thus P_{-2}^1(x) = -(1 - x^2)^{1/2}, demonstrating that the negative-degree functions replicate the positive ones without alteration. These examples illustrate the trivial equivalence for integer cases, underscoring their role primarily in theoretical extensions rather than new distinct polynomials.Fundamental Properties
Orthogonality Relations
The associated Legendre polynomials P_\ell^m(x) with fixed nonnegative integer order m \geq 0 and degrees \ell, \ell' \geq m satisfy an orthogonality relation over the interval [-1, 1] with respect to the constant weight function w(x) = 1: \int_{-1}^{1} P_\ell^m(x) P_{\ell'}^m(x) \, dx = \frac{2}{2\ell + 1} \frac{(\ell + m)!}{(\ell - m)!} \delta_{\ell \ell'}, where \delta_{\ell \ell'} is the Kronecker delta.[17] This relation implies that the polynomials are mutually orthogonal for distinct degrees \ell \neq \ell', with the nonzero integral providing the squared norm for each P_\ell^m(x). The proof of orthogonality follows from the associated Legendre differential equation, which is a Sturm-Liouville problem in self-adjoint form: \frac{d}{dx} \left[ (1 - x^2) \frac{d}{dx} P_\ell^m(x) \right] + \left[ \ell(\ell + 1) - \frac{m^2}{1 - x^2} \right] P_\ell^m(x) = 0. Consider two solutions u = P_\ell^m(x) and v = P_{\ell'}^m(x) satisfying analogous equations with eigenvalues \lambda_\ell = \ell(\ell + 1) and \lambda_{\ell'} = \ell'(\ell' + 1). Multiplying the equation for u by v and that for v by u, subtracting, and integrating over [-1, 1] yields (\lambda_\ell - \lambda_{\ell'}) \int_{-1}^{1} u v \, dx = \left[ v (1 - x^2) \frac{du}{dx} - u (1 - x^2) \frac{dv}{dx} \right]_{-1}^{1}. The boundary term vanishes because $1 - x^2 = 0 at the endpoints x = \pm 1. Thus, for \ell \neq \ell', the integral is zero.[19] The normalization constant arises when \ell = \ell', requiring evaluation of the integral using the Rodrigues representation P_\ell^m(x) = \frac{(-1)^m}{2^\ell \ell !} (1 - x^2)^{m/2} \frac{d^{ \ell + m }}{dx^{ \ell + m }} (x^2 - 1)^\ell. Substituting into the integral and applying integration by parts \ell + m times reduces it to a known beta-function integral or direct computation, yielding the factor \frac{2}{2\ell + 1} \frac{(\ell + m)!}{(\ell - m)!}.[17] For negative orders m < 0, the functions are defined via the relation P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x), with |m| \leq \ell. Orthogonality holds analogously for fixed negative m, as the proportionality constant adjusts the norm by its square: the integral becomes \frac{2}{2\ell + 1} \frac{(\ell - |m|)!}{(\ell + |m|)!} \delta_{\ell \ell'}. This extension preserves the separation for distinct \ell.Parity Characteristics
The associated Legendre polynomials P_\ell^m(x) for integers \ell \geq m \geq 0 possess a well-defined parity under the substitution x \to -x, given by the relation P_\ell^m(-x) = (-1)^{\ell + m} P_\ell^m(x). This property indicates that P_\ell^m(x) is an even function when \ell + m is even and an odd function when \ell + m is odd.[1][12] The parity arises from the standard definition via the Rodrigues formula, P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_\ell(x), where P_\ell(x) is the Legendre polynomial satisfying P_\ell(-x) = (-1)^\ell P_\ell(x). The factor (1 - x^2)^{m/2} is even in x, while the m-th derivative of P_\ell(x) inherits a parity sign of (-1)^{\ell + m} due to each differentiation flipping the parity sign of the preceding function. Combining these with the conventional phase factor (-1)^m yields the overall relation.[12][20] This parity characteristic simplifies analytical manipulations and numerical evaluations, particularly over symmetric intervals like [-1, 1], by allowing computations on [0, 1] and extension via the symmetry relation, thereby halving the required effort in applications involving even or odd extensions. For negative m, the relation P_\ell^{-m}(x) = (-1)^m \frac{(\ell - m)!}{(\ell + m)!} P_\ell^m(x) preserves the same parity up to the constant phase.[1]Computational Formulas
Recurrence Relations
Associated Legendre polynomials can be computed efficiently using recurrence relations, which provide a stable alternative to direct evaluation via differentiation from the standard definition. These relations are particularly useful for generating higher-degree or higher-order polynomials from lower ones. For fixed order m, the three-term recurrence relation in the degree l is (l - m + 1) P_{l+1}^m(x) = (2l + 1) x P_l^m(x) - (l + m) P_{l-1}^m(x). This relation holds for integer l \ge m \ge 0 and x \in [-1, 1], and it generalizes the corresponding recurrence for the ordinary Legendre polynomials (when m = 0).[21] To initiate the recursion for fixed m, the starting values are the ordinary Legendre polynomials P_l^0(x) for m = 0, computed via their own three-term recurrence (l + 1) P_{l+1}^0(x) = (2l + 1) x P_l^0(x) - l P_{l-1}^0(x), with initial conditions P_0^0(x) = 1 and P_1^0(x) = x. For m > 0, the recursion begins at l = m, using the seed value P_m^m(x) = (-1)^m (2m - 1)!! (1 - x^2)^{m/2}, where !! denotes the double factorial, and P_{m-1}^m(x) = 0.[21] Numerical stability is a key consideration in applying these recurrences. Forward recursion in the degree l (increasing from low to high l) is generally stable for |x| \le 1, but backward recursion (starting from a high degree and recursing downward) can improve accuracy for certain parameter regimes, particularly when evaluating at points near the endpoints x = \pm 1 or for high orders m. The choice depends on the specific range of l and m; detailed analysis shows that recurrences applied in the "stable direction" minimize error growth.[22]Gaunt's Formula
Gaunt's formula provides a linear expansion for the product of two associated Legendre polynomials P_\ell^m(x) and P_{\ell'}^{m'}(x) as a finite sum of associated Legendre polynomials P_{\ell''}^{m''}(x) with m'' = m + m', where the sum runs over \ell'' from |\ell - \ell'| to \ell + \ell' in steps of 2, subject to the triangular inequality and |m + m'| \leq \ell''.[23] The formula is expressed as P_\ell^m(x) P_{\ell'}^{m'}(x) = \sum_{\ell''} g(\ell m, \ell' m'; \ell'' (m+m')) \, P_{\ell''}^{m+m'}(x), where the Gaunt coefficients g serve as Clebsch-Gordan-like expansion coefficients that enforce angular momentum coupling rules.[23][24] This expansion was originally developed by J. A. Gaunt in 1929 within the context of atomic physics calculations for helium triplets, where it facilitated the evaluation of matrix elements involving spherical harmonics.[25] The explicit form of the Gaunt coefficient is g(\ell m, \ell' m'; \ell'' k) = (-1)^{m + m'} \sqrt{(2\ell + 1)(2\ell' + 1)(2\ell'' + 1)} \begin{pmatrix} \ell & \ell' & \ell'' \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \ell & \ell' & \ell'' \\ -m & -m' & k \end{pmatrix}, with k = m + m', using Wigner 3j symbols; an equivalent integral representation arises from the orthogonality of associated Legendre polynomials over [-1, 1].[23][24] In quantum mechanics, these coefficients are essential for expanding products in multipole interactions and deriving selection rules for transitions between states with definite angular momentum, such as in atomic spectra.[25][24]Explicit Representations
First Few Associated Legendre Functions
The first few associated Legendre polynomials P_\ell^m(x), for non-negative integers \ell and m with $0 \leq m \leq \ell, serve as fundamental examples and are typically computed using the Rodrigues formula P_\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_\ell(x), where P_\ell(x) is the Legendre polynomial of degree \ell.[1] These explicit forms verify the general definition and illustrate the polynomial nature for |x| \leq 1. The expressions up to \ell = 3 are listed in the following table, following the Condon-Shortley phase convention that includes the factor (-1)^m.[12]| \ell | m | P_\ell^m(x) |
|---|---|---|
| 0 | 0 | $1 |
| 1 | 0 | x |
| 1 | 1 | -\sqrt{1 - x^2} |
| 2 | 0 | \frac{1}{2} (3x^2 - 1) |
| 2 | 1 | -3x \sqrt{1 - x^2} |
| 2 | 2 | $3 (1 - x^2) |
| 3 | 0 | \frac{1}{2} (5x^3 - 3x) |
| 3 | 1 | -\frac{3}{2} (5x^2 - 1) \sqrt{1 - x^2} |
| 3 | 2 | $15x (1 - x^2) |
| 3 | 3 | -15 (1 - x^2)^{3/2} |