Associated Legendre polynomials

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
or equivalently
where the indices "ℓ" and "m" (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on [−1, 1] only if "ℓ" and "m" are integers with 0 ≤ "m" ≤ "ℓ", or with trivially equivalent negative values. When in addition "m" is even, the function is a polynomial. When "m" is zero and "ℓ" integer, these functions are identical to the Legendre polynomials. In general, when "ℓ" and "m" are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when "m" is odd. The fully general class of functions with arbitrary real or complex values of "ℓ" and "m" are Legendre functions. In that case the parameters are usually labelled with Greek letters.
The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
Definition for non-negative integer parameters ℓ and "m".
These functions are denoted formula_3, where the superscript indicates the order, and not a power of "P". Their most straightforward definition is in terms
of derivatives of ordinary Legendre polynomials ("m" ≥ 0)
The (−1)"m" factor in this formula is known as the Condon–Shortley phase. Some authors omit it. The functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters "ℓ" and "m" follows by differentiating "m" times the Legendre equation for "P""ℓ":
Moreover, since by Rodrigues' formula,
the "P" can be expressed in the form
This equation allows extension of the range of "m" to: −"ℓ" ≤ "m" ≤ "ℓ". The definitions of "P""ℓ"±"m", resulting from this expression by substitution of ±"m", are proportional. Indeed,
equate the coefficients of equal powers on the left and right hand side of 
then it follows that the proportionality constant is
so that 
Alternative notations.
The following alternative notations are also used in literature:
Closed Form.
The Associated Legendre Polynomial can also be written as:
with simple monomials and the generalized form of the binomial coefficient.
Orthogonality.
The associated Legendre polynomials are not mutually orthogonal in general. For example, formula_13 is not orthogonal to formula_14. However, some subsets are orthogonal. Assuming 0 ≤ "m" ≤ "ℓ", they satisfy the orthogonality condition for fixed "m":
Where δ"k", "ℓ" is the Kronecker delta.
Also, they satisfy the orthogonality condition for fixed "ℓ":
Negative "m" and/or negative ℓ.
The differential equation is clearly invariant under a change in sign of "m".
The functions for negative "m" were shown above to be proportional to those of positive "m":
formula_18
The differential equation is also invariant under a change from "ℓ" to
−"ℓ" − 1, and the functions for negative "ℓ" are defined by
Parity.
From their definition, one can verify that the Associated Legendre functions are either even or odd according to
The first few associated Legendre functions.
The first few associated Legendre functions, including those for negative values of "m", are:
Recurrence formula.
These functions have a number of recurrence properties:
Helpful identities (initial values for the first recursion):
with !! the double factorial.
Gaunt's formula.
The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula 
This formula is to be used under the following assumptions: 
Other quantities appearing in the formula are defined as
The integral is zero unless
Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.
Generalization via hypergeometric functions.
These functions may actually be defined for general complex parameters and argument:
where formula_74 is the gamma function and formula_75 is the hypergeometric function
They are called the Legendre functions when defined in this more general way. They satisfy
the same differential equation as before:
Since this is a second order differential equation, it has a second solution, 
formula_78, defined as:
formula_80 and formula_78 both obey the various
recurrence formulas given previously.
Reparameterization in terms of angles.
These functions are most useful when the argument is reparameterized in terms of angles,
letting formula_82:
Using the relation formula_84, the list given above yields the first few polynomials, parameterized this way, as:
The orthogonality relations given above become in this formulation:
for fixed "m", formula_86 are orthogonal, parameterized by θ over formula_87, with weight formula_88:
Also, for fixed "ℓ":
In terms of θ, formula_91 are solutions of
More precisely, given an integer "m"formula_930, the above equation has
nonsingular solutions only when formula_94 for "ℓ"
an integer ≥ "m", and those solutions are proportional to
formula_91.
Applications in physics: spherical harmonics.
In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved. The colatitude angle in spherical coordinates is
the angle formula_96 used above. The longitude angle, formula_97, appears in a multiplying factor. Together, they make a set of functions called spherical harmonics. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3).
What makes these functions useful is that they are central to the solution of the equation
formula_98 on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is
When the partial differential equation
is solved by the method of separation of variables, one gets a φ-dependent part formula_101 or formula_102 for integer m≥0, and an equation for the θ-dependent part
for which the solutions are formula_91 with formula_105
and formula_106.
Therefore, the equation
has nonsingular separated solutions only when formula_106,
and those solutions are proportional to
and
For each choice of "ℓ", there are functions
for the various values of "m" and choices of sine and cosine.
They are all orthogonal in both "ℓ" and "m" when integrated over the
surface of the sphere.
The solutions are usually written in terms of complex exponentials:
The functions formula_112 are the spherical harmonics, and the quantity in the square root is a normalizing factor.
Recalling the relation between the associated Legendre functions of positive and negative "m", it is easily shown that the spherical harmonics satisfy the identity
The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics).
When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically
of the form
and hence the solutions are spherical harmonics.
Generalizations.
The Legendre polynomials are closely related to hypergeometric series. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3). There are many other Lie groups besides SO(3), and an analogous generalization of the Legendre polynomials exist to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces. Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings.