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 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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
. 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
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s when ''m'' is odd. The fully general class of functions with arbitrary real or complex values of ''ℓ'' and ''m'' are
Legendre function
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ...
s. In that case the parameters are usually labelled with Greek letters.
The Legendre
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
is frequently encountered in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
and other technical fields. In particular, it occurs when solving
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
(and related
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
s) in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' mea ...
. Associated Legendre polynomials play a vital role in the definition of
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
.
Definition for non-negative integer parameters and
These functions are denoted
, 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
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
(''m'' ≥ 0)
The factor in this formula is known as the
Condon–Shortley phase. Some authors omit it. That 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 :
Moreover, since by
Rodrigues' formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out ...
,
the ''P'' can be expressed in the form
This equation allows extension of the range of ''m'' to: . The definitions of , resulting from this expression by substitution of , 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,
is not orthogonal to
. However, some subsets are orthogonal. Assuming 0 ≤ ''m'' ≤ ''ℓ'', they satisfy the orthogonality condition for fixed ''m'':
Where is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 ...
.
Also, they satisfy the orthogonality condition for fixed :
Negative and/or negative
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'':
(This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative .)
The differential equation is also invariant under a change from to , and the functions for negative 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: