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Associated Legendre Polynomial

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, where l is a positive integer and m=0, ..., l. They are implemented in the Wolfram Language as LegendreP[l, m, x]. For positive m, they can be given in terms of the unassociated polynomials by

P_l^m(x)=(-1)^m(1-x^2)^(m/2)(d^m)/(dx^m)P_l(x)
(1)
=((-1)^m)/(2^ll!)(1-x^2)^(m/2)(d^(l+m))/(dx^(l+m))(x^2-1)^l,
(2)

where P_l(x) are the unassociated Legendre polynomials. The associated Legendre polynomials for negative m are then defined by

P_l^(-m)(x)=(-1)^m((l-m)!)/((l+m)!)P_l^m(x).
(3)

There are two sign conventions for associated Legendre polynomials. Some authors (e.g., Arfken 1985, pp. 668-669) omit the Condon-Shortley phase (-1)^m, while others include it (e.g., Abramowitz and Stegun 1972, Press et al. 1992, and the LegendreP[l, m, z] command in the Wolfram Language). Care is therefore needed in comparing polynomials obtained from different sources. One possible way to distinguish the two conventions is due to Abramowitz and Stegun (1972, p. 332), who use the notation

P_(lm)(x)=(-1)^mP_l^m(x)
(4)

to distinguish the two.

Associated polynomials are sometimes called Ferrers' functions (Sansone 1991, p. 246). If m=0, they reduce to the unassociated polynomials. The associated Legendre functions are part of the spherical harmonics, which are the solution of Laplace's equation in spherical coordinates. They are orthogonal over [-1,1] with the weighting function 1

int_(-1)^1P_l^m(x)P_(l^')^m(x)dx=2/(2l+1)((l+m)!)/((l-m)!)delta_(ll^'),
(5)

and orthogonal over [-1,1] with respect to m with the weighting function (1-x^2)^(-1),

int_(-1)^1P_l^m(x)P_l^(m^')(x)(dx)/(1-x^2)=((l+m)!)/(m(l-m)!)delta_(mm^').
(6)

The associated Legendre polynomials also obey the following recurrence relations

(l-m)P_l^m(x)=x(2l-1)P_(l-1)^m(x)-(l+m-1)P_(l-2)^m(x).
(7)

Letting x=costheta (commonly denoted mu in this context),

(dP_l^m(mu))/(dtheta)=(lmuP_l^m(mu)-(l+m)P_(l-1)^m(mu))/(sqrt(1-mu^2))
(8)
(2l+1)muP_l^m(mu)=(l+m)P_(l-1)^m(mu)+(l-m+1)P_(l+1)^m(mu).
(9)

Additional identities are

P_l^l(x)=(-1)^l(2l-1)!!(1-x^2)^(l/2)
(10)
P_(l+1)^l(x)=x(2l+1)P_l^l(x).
(11)

Including the factor of (-1)^m, the first few associated Legendre polynomials are

P_0^0(x)=1
(12)
P_1^0(x)=x
(13)
P_1^1(x)=-(1-x^2)^(1/2)
(14)
P_2^0(x)=1/2(3x^2-1)
(15)
P_2^1(x)=-3x(1-x^2)^(1/2)
(16)
P_2^2(x)=3(1-x^2)
(17)
P_3^0(x)=1/2x(5x^2-3)
(18)
P_3^1(x)=3/2(1-5x^2)(1-x^2)^(1/2)
(19)
P_3^2(x)=15x(1-x^2)
(20)
P_3^3(x)=-15(1-x^2)^(3/2)
(21)
P_4^0(x)=1/8(35x^4-30x^2+3)
(22)
P_4^1(x)=5/2x(3-7x^2)(1-x^2)^(1/2)
(23)
P_4^2(x)=(15)/2(7x^2-1)(1-x^2)
(24)
P_4^3(x)=-105x(1-x^2)^(3/2)
(25)
P_4^4(x)=105(1-x^2)^2
(26)
P_5^0(x)=1/8x(63x^4-70x^2+15).
(27)

Written in terms x=costheta (commonly written mu=costheta), the first few become

P_0^0(costheta)=1
(28)
P_1^0(costheta)=costheta
(29)
P_1^1(costheta)=-sintheta
(30)
P_2^0(costheta)=1/2(3cos^2theta-1)
(31)
P_2^1(costheta)=-3sinthetacostheta
(32)
P_2^2(costheta)=3sin^2theta
(33)
P_3^0(costheta)=1/2costheta(5cos^2theta-3)
(34)
P_3^1(costheta)=-3/2(5cos^2theta-1)sintheta
(35)
P_3^2(costheta)=15costhetasin^2theta
(36)
P_3^3(costheta)=-15sin^3theta.
(37)

The derivative about the origin is

[(dP_nu^mu(x))/(dx)]_(x=0)=(2^(mu+1)sin[1/2pi(nu+mu)]Gamma(1/2nu+1/2mu+1))/(pi^(1/2)Gamma(1/2nu-1/2mu+1/2))
(38)

(Abramowitz and Stegun 1972, p. 334), and the logarithmic derivative is

[(dlnP_lambda^mu(z))/(dz)]_(z=0)=2tan[1/2pi(lambda+mu)]([1/2(lambda+mu)]![1/2(lambda-mu)]!)/([1/2(lambda+mu-1)]![1/2(lambda-mu-1)]!).
(39)

(Binney and Tremaine 1987, p. 654).

See also

Associated Legendre Differential Equation, Condon-Shortley Phase, Gegenbauer Polynomial, Legendre Function of the First Kind, Legendre Function of the Second Kind, Legendre Polynomial, Spherical Harmonic, Toroidal Function

Related Wolfram sites

http://functions.wolfram.com/Polynomials/LegendreP2/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" and "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972.Arfken, G. "Legendre Functions." Ch. 12 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 637-711, 1985.Bailey, W. N. "On the Product of Two Legendre Polynomials." Proc. Cambridge Philos. Soc. 29, 173-177, 1933.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Byerly, W. E. "Zonal Harmonics." Ch. 5 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 144-194, 1959.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, 1956.Iyanaga, S. and Kawada, Y. (Eds.). "Legendre Function" and "Associated Legendre Function." Appendix A, Tables 18.II and 18.III in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1462-1468, 1980.Koekoek, R. and Swarttouw, R. F. "Legendre / Spherical." §1.8.3 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, p. 44, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Lagrange, R. Polynomes et fonctions de Legendre. Paris: Gauthier-Villars, 1939.Legendre, A. M. "Sur l'attraction des Sphéroides." Mém. Math. et Phys. présentés à l'Ac. r. des. sc. par divers savants 10, 1785.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 593-597, 1953.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 252, 1992.Sansone, G. "Expansions in Series of Legendre Polynomials and Spherical Harmonics." Ch. 3 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 169-294, 1991.Sloane, N. J. A. Sequences A001790/M2508, A002596/M3768, A008316, A008317, A046161, A060818, A078297, and A078298 in "The On-Line Encyclopedia of Integer Sequences."Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952.Spanier, J. and Oldham, K. B. "The Legendre Polynomials P_n(x)" and "The Legendre Functions P_nu(x) and Q_nu(x)." Chs. 21 and 59 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 183-192 and 581-597, 1987.Strutt, J. W. "On the Values of the Integral int_0^1Q_nQ_n^'dmu, Q_n, Q_n^' being LaPlace's Coefficients of the orders n, n^', with an Application to the Theory of Radiation." Philos. Trans. Roy. Soc. London 160, 579-590, 1870.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Associated Legendre Polynomial

Cite this as:

Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AssociatedLegendrePolynomial.html

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