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

where
are the unassociated Legendre polynomials. The associated Legendre polynomials for
negative
are then defined by

(3)

There are two sign conventions for associated Legendre polynomials. Some authors (e.g., Arfken 1985, pp. 668-669) omit the Condon-Shortley
phase , 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

(4)

to distinguish the two.

Associated polynomials are sometimes called Ferrers' functions (Sansone 1991, p. 246). If ,
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
with the weighting function 1

(5)

and orthogonal over with respect to with the weighting function ,

(6)

The associated Legendre polynomials also obey the following recurrence
relations

(7)

Letting
(commonly denoted in this context),

(8)

(9)

Additional identities are

(10)

(11)

Including the factor of , the first few associated Legendre polynomials are

Written in terms (commonly written ), the first few become

The derivative about the origin is

(38)

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

(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/
Explore with Wolfram|Alpha
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
-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 " and "The Legendre Functions and ." 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 , , being LaPlace's Coefficients of the orders , , 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. Referenced on Wolfram|Alpha 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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