{{Short description|Canonical solutions of the general Legendre equation}}In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equationleft(1 - x^2right) frac{d^2}{d x^2} P_ell^m(x) - 2 x frac{d}{d x} P_ell^m(x) + left[ ell (ell + 1) - frac{m^2}{1 - x^2} right] P_ell^m(x) = 0,or equivalentlyfrac{d}{d x} left[ left(1 - x^2right) frac{d}{d x} P_ell^m(x) right] + left[ ell (ell + 1) - frac{m^2}{1 - x^2} right] P_ell^m(x) = 0,where the indices â„“ 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 {{closed-closed|−1, 1}} only if â„“ and m are integers with 0 ≤ m ≤ â„“, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and â„“ integer, these functions are identical to the Legendre polynomials. In general, when â„“ 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 â„“ 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 {{mvar|â„“}} and {{mvar|m}}

These functions are denoted P_ell^{m}(x), where the superscript indicates the order and not a power of P. Their most straightforward definition is in termsof derivatives of ordinary Legendre polynomials (m ≥ 0) The {{math|(−1)m}} 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 â„“ and m follows by differentiating m times the Legendre equation for {{math|Pâ„“}}:{{harvnb|Courant|Hilbert|1953|loc=V, §10}}.left(1-x^2right) frac{d^2}{dx^2}P_ell(x) -2xfrac{d}{dx}P_ell(x)+ ell(ell+1)P_ell(x) = 0.Moreover, since by Rodrigues’ formula,P_ell(x) = frac{1}{2^ell,ell!} frac{d^ell}{dx^ell}left[(x^2-1)^ellright],the P{{su|b=â„“|p=m}} can be expressed in the formP_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.This equation allows extension of the range of m to: {{math|−ℓ ≤ m ≤ â„“}}. The definitions of {{math|Pℓ±m}}, resulting from this expression by substitution of {{math|±m}}, are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side offrac{d^{ell-m}}{dx^{ell-m}} (x^2-1)^{ell} = c_{lm} (1-x^2)^m frac{d^{ell+m}}{dx^{ell+m}}(x^2-1)^{ell},then it follows that the proportionality constant isc_{lm} = (-1)^m frac{(ell-m)!}{(ell+m)!} ,so thatP^{-m}_ell(x) = (-1)^m frac{(ell-m)!}{(ell+m)!} P^{m}_ell(x).

Alternative notations

The following alternative notations are also used in literature:{{Abramowitz_Stegun_ref|8|332}}P_{ell m}(x) = (-1)^m P_ell^{m}(x)

Closed Form

The Associated Legendre Polynomial can also be written as:{{Citation needed|reason=This appears correct, at least for positive m, but is non-obvious.|date=March 2024}} with simple monomials and the generalized form of the binomial coefficient.

Orthogonality

The associated Legendre polynomials are not mutually orthogonal in general. For example, P_1^1 is not orthogonal to P_2^2. However, some subsets are orthogonal. Assuming 0 â‰¤ m â‰¤ â„“, they satisfy the orthogonality condition for fixed m:int_{-1}^{1} P_k ^{m} P_ell ^{m} dx = frac{2 (ell+m)!}{(2ell+1)(ell-m)!} delta _{k,ell}Where {{math|δk,â„“}} is the Kronecker delta.Also, they satisfy the orthogonality condition for fixed {{mvar|â„“}}:int_{-1}^{1} frac{P_ell ^{m} P_ell ^{n}}{1-x^2}dx = begin{cases}frac{(ell+m)!}{m(ell-m)!} & text{if } m=nneq0 infty & text{if } m=n=0end{cases}

Negative {{mvar|m}} and/or negative {{mvar|â„“}}

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:P_ell ^{-m} = (-1)^m frac{(ell-m)!}{(ell+m)!} P_ell ^{m}(This followed from the Rodrigues’ formula definition. This definition also makes the various recurrence formulas work for positive or negative {{mvar|m}}.)text{If}quad |m| > ell,quadtext{then}quad P_ell^{m} = 0.,The differential equation is also invariant under a change from {{mvar|â„“}} to {{math|−ℓ − 1}}, and the functions for negative {{mvar|â„“}} are defined byP_{-ell} ^{m} = P_{ell-1} ^{m}, (ell=1,,2,, dots).

Parity

From their definition, one can verify that the Associated Legendre functions are either even or odd according toP_ell ^{m} (-x) = (-1)^{ell - m} P_ell ^{m}(x)

The first few associated Legendre functions

(File:Mplwp legendreP04a0.svg|thumb|300px|Associated Legendre functions for m = 0)(File:Mplwp legendreP15a1.svg|thumb|300px|Associated Legendre functions for m = 1)(File:Mplwp legendreP26a2.svg|thumb|300px|Associated Legendre functions for m = 2)The first few associated Legendre functions, including those for negative values of m, are:P_{0}^{0}(x)=1begin{align}P_{1}^{-1}(x)&=-tfrac{1}{2}P_{1}^{1}(x) P_{1}^{0}(x)&=x P_{1}^{1}(x)&=-(1-x^2)^{1/2}end{align}begin{align}P_{2}^{-2}(x)&=tfrac{1}{24}P_{2}^{2}(x) P_{2}^{-1}(x)&=-tfrac{1}{6}P_{2}^{1}(x) P_{2}^{0}(x)&=tfrac{1}{2}(3x^{2}-1) P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} P_{2}^{2}(x)&=3(1-x^2)end{align}begin{align}P_{3}^{-3}(x)&=-tfrac{1}{720}P_{3}^{3}(x) P_{3}^{-2}(x)&=tfrac{1}{120}P_{3}^{2}(x) P_{3}^{-1}(x)&=-tfrac{1}{12}P_{3}^{1}(x) P_{3}^{0}(x)&=tfrac{1}{2}(5x^3-3x) P_{3}^{1}(x)&=tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} P_{3}^{2}(x)&=15x(1-x^2) P_{3}^{3}(x)&=-15(1-x^2)^{3/2}end{align}begin{align}P_{4}^{-4}(x)&=tfrac{1}{40320}P_{4}^{4}(x) P_{4}^{-3}(x)&=-tfrac{1}{5040}P_{4}^{3}(x) P_{4}^{-2}(x)&=tfrac{1}{360}P_{4}^{2}(x) P_{4}^{-1}(x)&=-tfrac{1}{20}P_{4}^{1}(x) P_{4}^{0}(x)&=tfrac{1}{8}(35x^{4}-30x^{2}+3) P_{4}^{1}(x)&=-tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} P_{4}^{2}(x)&=tfrac{15}{2}(7x^2-1)(1-x^2) P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} P_{4}^{4}(x)&=105(1-x^2)^{2}end{align}

Recurrence formula

These functions have a number of recurrence properties:(ell-m-1)(ell-m)P_{ell}^{m}(x) = -P_{ell}^{m+2}(x) + P_{ell-2}^{m+2}(x) + (ell+m)(ell+m-1)P_{ell-2}^{m}(x)(ell-m+1)P_{ell+1}^{m}(x) = (2ell+1)xP_{ell}^{m}(x) - (ell+m)P_{ell-1}^{m}(x)2mxP_{ell}^{m}(x)=-sqrt{1-x^2}left[P_{ell}^{m+1}(x)+(ell+m)(ell-m+1)P_{ell}^{m-1}(x)right]frac{1}{sqrt{1-x^2}}P_ell^m(x) = frac{-1}{2m} left[ P_{ell-1}^{m+1}(x) + (ell+m-1)(ell+m)P_{ell-1}^{m-1}(x) right]frac{1}{sqrt{1-x^2}}P_ell^m(x) = frac{-1}{2m} left[ P_{ell+1}^{m+1}(x) + (ell-m+1)(ell-m+2)P_{ell+1}^{m-1}(x) right]

sqrt{1-x^2}P_ell^{m+1}(x) = (ell-m)xP_{ell}^{m}(x) - (ell+m)P_{ell-1}^{m}(x)sqrt{1-x^2}P_ell^{m+1}(x) = (ell-m+1)P_{ell+1}^m(x) - (ell+m+1)xP_ell^m(x)

(x^2-1)frac{d}{dx}{P_{ell}^{m}}(x) = {ell}xP_{ell}^{m}(x) - (ell+m)P_{ell-1}^{m}(x)(x^2-1)frac{d}{dx}{P_{ell}^{m}}(x) = -(ell+1)xP_{ell}^{m}(x) + (ell-m+1)P_{ell+1}^{m}(x)(x^2-1)frac{d}{dx}{P_{ell}^{m}}(x) = sqrt{1-x^2}P_{ell}^{m+1}(x) + mxP_{ell}^{m}(x)(x^2-1)frac{d}{dx}{P_{ell}^{m}}(x) = -(ell+m)(ell-m+1)sqrt{1-x^2}P_{ell}^{m-1}(x) - mxP_{ell}^{m}(x)Helpful identities (initial values for the first recursion):P_{ell +1}^{ell +1}(x) = - (2ell+1) sqrt{1-x^2} P_{ell}^{ell}(x)P_{ell}^{ell}(x) = (-1)^ell (2ell-1)!! (1- x^2)^{(ell/2)}P_{ell +1}^{ell}(x) = x (2ell+1) P_{ell}^{ell}(x)with {{math|!!}} 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 From John C. Slater Quantum Theory of Atomic Structure, McGraw-Hill (New York, 1960), Volume I, page 309, which cites the original work of J. A. Gaunt, Philosophical Transactions of the Royal Society of London, A228:151 (1929)begin{align}frac{1}{2} int_{-1}^1 P_l^u(x) P_m^v(x) P_n^w(x) dx

{}&{}(-1)^{s-m-w}frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!}

end{align}This formula is to be used under the following assumptions:

1. the degrees are non-negative integers l,m,nge0
2. all three orders are non-negative integers u,v,wge 0
3. u is the largest of the three orders
4. the orders sum up u=v+w
5. the degrees obey mge n

Other quantities appearing in the formula are defined as The integral is zero unless

1. the sum of degrees is even so that s is an integer
2. the triangular condition is satisfied m+nge l ge m-n

Dong and Lemus (2002)Dong S.H., Lemus R., (2002), “The overlap integral of three associated Legendre polynomials”, Appl. Math. Lett. 15, 541-546. 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:P_{lambda}^{mu}(z) = frac{1}{Gamma(1-mu)} left[frac{1+z}{1-z}right]^{mu/2} ,_2F_1 (-lambda, lambda+1; 1-mu; frac{1-z}{2})where Gamma is the gamma function and _2F_1 is the hypergeometric function,_2F_1 (alpha, beta; gamma; z) = frac{Gamma(gamma)}{Gamma(alpha)Gamma(beta)} sum_{n=0}^inftyfrac{Gamma(n+alpha)Gamma(n+beta)}{Gamma(n+gamma) n!}z^n,They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:(1-z^2),y’’ -2zy’ + left(lambda[lambda+1] - frac{mu^2}{1-z^2}right),y = 0.,Since this is a second order differential equation, it has a second solution, Q_lambda^{mu}(z), defined as:Q_{lambda}^{mu}(z) = frac{sqrt{pi} Gamma(lambda+mu+1)}{2^{lambda+1}Gamma(lambda+3/2)}frac{1}{z^{lambda+mu+1}}(1-z^2)^{mu/2} ,_2F_1 left(frac{lambda+mu+1}{2}, frac{lambda+mu+2}{2}; lambda+frac{3}{2}; frac{1}{z^2}right)P_lambda^{mu}(z) and Q_lambda^{mu}(z) 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 x = costheta:P_ell^{m}(costheta) = (-1)^m (sin theta)^m frac{d^m}{d(costheta)^m}left(P_ell(costheta)right)Using the relation (1 - x^2)^{1 / 2} = sintheta, the list given above yields the first few polynomials, parameterized this way, as:begin{align}P_0^0(costheta) & = 1 [8pt]P_1^0(costheta) & = costheta [8pt]P_1^1(costheta) & = -sintheta [8pt]P_2^0(costheta) & = tfrac{1}{2} (3cos^2theta-1) [8pt]P_2^1(costheta) & = -3costhetasintheta [8pt]P_2^2(costheta) & = 3sin^2theta [8pt]P_3^0(costheta) & = tfrac{1}{2} (5cos^3theta-3costheta) [8pt]P_3^1(costheta) & = -tfrac{3}{2} (5cos^2theta-1)sintheta [8pt]P_3^2(costheta) & = 15costhetasin^2theta [8pt]P_3^3(costheta) & = -15sin^3theta [8pt]P_4^0(costheta) & = tfrac{1}{8} (35cos^4theta-30cos^2theta+3) [8pt]P_4^1(costheta) & = - tfrac{5}{2} (7cos^3theta-3costheta)sintheta [8pt]P_4^2(costheta) & = tfrac{15}{2} (7cos^2theta-1)sin^2theta [8pt]P_4^3(costheta) & = -105costhetasin^3theta [8pt]P_4^4(costheta) & = 105sin^4thetaend{align}The orthogonality relations given above become in this formulation:for fixed m, P_ell^m(costheta) are orthogonal, parameterized by θ over [0, pi], with weight sin theta:int_0^pi P_k^{m}(costheta) P_ell^{m}(costheta),sintheta,dtheta = frac{2 (ell+m)!}{(2ell+1)(ell-m)!} delta _{k,ell}Also, for fixed â„“:int_0^pi P_ell^{m}(costheta) P_ell^{n}(costheta) csctheta,dtheta = begin{cases} 0 & text{if } mneq n frac{(ell+m)!}{m(ell-m)!} & text{if } m=nneq0 infty & text{if } m=n=0end{cases}In terms of θ, P_ell^{m}(cos theta) are solutions offrac{d^{2}y}{dtheta^2} + cot theta frac{dy}{dtheta} + left[lambda - frac{m^2}{sin^2theta}right],y = 0,More precisely, given an integer mge0, the above equation hasnonsingular solutions only when lambda = ell(ell+1), for â„“an integer ≥ m, and those solutions are proportional toP_ell^{m}(cos theta).

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 isthe angle theta used above. The longitude angle, phi, 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).{{cn|date=July 2022}}What makes these functions useful is that they are central to the solution of the equationnabla^2psi + lambdapsi = 0 on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian isnabla^2psi = frac{partial^2psi}{partialtheta^2} + cot theta frac{partial psi}{partial theta} + csc^2 thetafrac{partial^2psi}{partialphi^2}.When the partial differential equationfrac{partial^2psi}{partialtheta^2} + cot theta frac{partial psi}{partial theta} + csc^2 thetafrac{partial^2psi}{partialphi^2} + lambda psi = 0is solved by the method of separation of variables, one gets a φ-dependent part sin(mphi) or cos(mphi) for integer m≥0, and an equation for the θ-dependent partfrac{d^{2}y}{dtheta^2} + cot theta frac{dy}{dtheta} + left[lambda - frac{m^2}{sin^2theta}right],y = 0,for which the solutions are P_ell^{m}(cos theta) with ell{ge}mand lambda = ell(ell+1).Therefore, the equationnabla^2psi + lambdapsi = 0has nonsingular separated solutions only when lambda = ell(ell+1),and those solutions are proportional toP_ell^{m}(cos theta) cos (mphi) 0 le m le ellandP_ell^{m}(cos theta) sin (mphi) 0 < m le ell.For each choice of ℓ, there are {{nowrap|2ℓ + 1}} functionsfor the various values of m and choices of sine and cosine.They are all orthogonal in both ℓ and m when integrated over thesurface of the sphere.The solutions are usually written in terms of complex exponentials:Y_{ell, m}(theta, phi) = sqrt{frac{(2ell+1)(ell-m)!}{4pi(ell+m)!}} P_ell^{m}(cos theta) e^{imphi}qquad -ell le m le ell.The functions Y_{ell, m}(theta, phi) 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 identityThis identity can also be shown by relating the spherical harmonics to Wigner D-matrices and use of the time-reversal property of the latter. The relation between associated Legendre functions of ±m can then be proved from the complex conjugation identity of the spherical harmonics.Y_{ell, m}^*(theta, phi) = (-1)^m Y_{ell, -m}(theta, phi).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 typicallyof the formnabla^2psi(theta, phi) + lambdapsi(theta, phi) = 0,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 analogous generalizations 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.

See also

• Angular momentum
• Gaussian quadrature
• Legendre polynomials
• Spherical harmonics
• Whipple’s transformation of Legendre functions
• Laguerre polynomials
• Hermite polynomials

Notes and references

{{Reflist}}

• {{citation|last1=Arfken|first1=G.B.|last2=Weber|first2=H.J.|title=Mathematical methods for physicists|year=2001|publisher=Academic Press|isbn=978-0-12-059825-0}}; Section 12.5. (Uses a different sign convention.)
• {{citation|last=Belousov|first=S. L.|year=1962|title=Tables of normalized associated Legendre polynomials|series=Mathematical tables|volume=18|publisher=Pergamon Press}}.
• {{citation|first1=E. U.|last1=Condon|first2=G. H.|last2=Shortley|title=The Theory of Atomic Spectra|year=1970|location=Cambridge, England|publisher=Cambridge University Press|oclc=5388084}}; Chapter 3.
• {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publischer, Inc|location=New York}}.
• {{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
• {{citation|first=A.R.|last=Edmonds|title=Angular Momentum in Quantum Mechanics|year=1957|publisher=Princeton University Press|isbn=978-0-691-07912-7|url-access=registration|url=https://archive.org/details/angularmomentumi0000edmo}}; Chapter 2.
• {{citation|first=F. B.|last=Hildebrand|authorlink=Francis B. Hildebrand|title=Advanced Calculus for Applications|year=1976|publisher=Prentice Hall|isbn=978-0-13-011189-0}}.
• {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
• Schach, S. R. (1973) New Identities for Legendre Associated Functions of Integral Order and Degree , Society for Industrial and Applied Mathematics Journal on Mathematical Analysis, 1976, Vol. 7, No. 1 : pp. 59–69

External links

• Associated Legendre polynomials in MathWorld
• Legendre polynomials in MathWorld
• Legendre and Related Functions in DLMF

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