Legendre Polynomial
The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients,
or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to the Legendre differential
equation. If is an integer, they are polynomials. The Legendre polynomials
for
and
The Legendre polynomial
are illustrated above
, 2, ..., 5. They are implemented in Mathematica as LegendreP[n, x].
can be defined by the contour integral
(1)
where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985,
p. 416).
The first few Legendre polynomials are
(2)
(3)
(4)
(5)
(6)
(7)
(8)
When ordered from smallest to largest powers and with the denominators factored out, the triangle of
nonzero coefficients is 1, 1,
, 3,
, 5, 3,
, ... (Sloane's A008316). The leading denominators are
1, 1, 2, 2, 8, 8, 16, 16, 128, 128, 256, 256, ... (Sloane's A060818).
The first few powers in terms of Legendre polynomials are
(9)
(10)
(11)
(12)
(13)
(14)
(Sloane's A008317 and A001790). A closed form for these is given by
(15)
(R. Schmied, pers. comm., Feb. 27, 2005). For Legendre polynomials and powers up to exponent 12,
see Abramowitz and Stegun (1972, p. 798).
The Legendre polynomials can also be generated using Gram-Schmidt orthonormalization in the open
interval
with the weighting function 1.
(16)
(17)
(18)
(19)
(20)
(21)
(22)
Normalizing so that
gives the expected Legendre polynomials.
The "shifted" Legendre polynomials are a set of functions analogous to the Legendre polynomials, but
defined on the interval (0, 1). They obey the orthogonality relationship
(23)
The first few are
(24)
(25)
(26)
(27)
The Legendre polynomials are orthogonal over
with weighting function 1 and satisfy
(28)
where
is the Kronecker delta.
The Legendre polynomials are a special case of the ultraspherical functions with
case of the Jacobi polynomials
with
, a special
, and can be written as a hypergeometric function
using Murphy's formula
(29)
(Bailey 1933; 1935, p. 101; Koekoek and Swarttouw 1998).
The Rodrigues representation provides the formula
(30)
which yields upon expansion
(31)
(32)
where
is the floor function. Additional sum formulas include
(33)
(34)
(Koepf 1998, p. 1). In terms of hypergeometric functions, these can be written
(35)
(36)
(37)
(Koepf 1998, p. 3).
A generating function for
is given by
(38)
Take
,
(39)
Multiply (39) by
,
(40)
and add (38) and (40),
(41)
This expansion is useful in some physical problems, including expanding the Heyney-Greenstein phase
function and computing the charge distribution on a sphere. Another generating function is given by
(42)
where
is a zeroth order Bessel function of the first kind (Koepf 1998, p. 2).
The Legendre polynomials satisfy the recurrence relation
(43)
(Koepf 1998, p. 2). In addition,
(44)
(correcting Hildebrand 1956, p. 324).
A complex generating function is
(45)
and the Schläfli integral is
(46)
Integrals over the interval
include the general formula
(47)
for
(Byerly 1959, p. 172), from which the special case
(48)
(49)
follows (Sloane's A002596 and A046161; Byerly 1959, p. 172). For the integral over a product of
Legendre functions,
(50)
for
(Byerly 1959, p. 172), which gives the special case
(51)
where
(52)
(Sloane's A078297 and A078298; Byerly 1959, p. 172). The latter is a special case of
(53)
where
(54)
and
is a gamma function (Gradshteyn and Ryzhik 2000, p. 762, eqn. 7.113.1)
Integrals over
with weighting functions and
are given by
(55
)
(56
)
(Arfken 1985, p. 700).
The Laplace transform is given by
(57
)
where
is a modified Bessel function of the first kind.
A sum identity is given by
(58)
where
is the th root of
(Szegö 1975, p. 348). A similar identity is
(59)
which is responsible for the fact that the sum of weights in Legendre-Gauss quadrature is always equal
to 2.
The associated Legendre polynomials
and
differential equation, where is a positive integer and
as LegendreP[l, m, x]. For positive
are solutions to the associated Legendre
, ..., . They are implemented in Mathematica
, they can be given in terms of the unassociated polynomials by
(60)
(61)
where
negative
are the unassociated Legendre polynomials. The associated Legendre polynomials for
are then defined by
(62)
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 of Mathematica). 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
(63)
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
(64)
and orthogonal over
with respect to
with the weighting function
,
(65)
The associated Legendre polynomials also obey the following recurrence relations
(66)
Letting
(commonly denoted in this context),
(67)
(68)
Additional identities are
(69)
(70)
Including the factor of
, the first few associated Legendre polynomials are
(71)
(72)
(73)
(74)
(75)
(76)
(77)
(78)
(79)
(80)
(81)
(82)
(83)
(84)
(85)
(86)
Written in terms
(commonly written
), the first few become
(87)
(88)
(89)
(90)
(91)
(92)
(93)
(94)
(95)
(96)
The derivative about the origin is
(97)
(Abramowitz and Stegun 1972, p. 334), and the logarithmic derivative is
(98)
(Binney and Tremaine 1987, p. 654).
SEE ALSO: Condon-Shortley Phase, Conical Function, Kings Problem, Laplace's Integral, Laplace-
Mehler Integral, Legendre Function of the First Kind, Legendre Function of the Second Kind, MehlerDirichlet Integral, Spherical Harmonic, Super Catalan Number, Toroidal Function, Turán's Inequalities,
Ultraspherical Polynomial, Zonal Harmonic
RELATED WOLFRAM SITES: http://functions.wolfram.com/Polynomials/LegendreP/,
http://functions.wolfram.com/Polynomials/LegendreP/
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.
Binney, J. and Tremaine, S. "Associated Legendre Functions." Appendix 5 in Galactic Dynamics. Princeton, NJ: Princeton
University Press, pp. 654-655, 1987.
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.