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Legendre Polynomial

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LegendreP

The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are the polynomial solutions P_n of the Legendre differential equation, normalized by P_n(1)=1, for nonnegative integers n. The Legendre polynomials P_n(x) are illustrated above for x in [-1,1] and n=1, 2, ..., 5. They are implemented in the Wolfram Language as LegendreP[n, x].

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) are solutions to the associated Legendre differential equation, where l is a nonnegative integer and m=0, ..., l.

The Legendre polynomial P_n(z) can be defined by the contour integral

P_n(z)=1/(2pii)∮(1-2tz+t^2)^(-1/2)t^(-n-1)dt,
(1)

where n is a nonnegative integer and the contour is a sufficiently small counterclockwise circle about the origin. The square root is the analytic function branch taking the value 1 at t=0 (Arfken 1985, p. 416).

The first few Legendre polynomials are

P_0(x)=1
(2)
P_1(x)=x
(3)
P_2(x)=1/2(3x^2-1)
(4)
P_3(x)=1/2(5x^3-3x)
(5)
P_4(x)=1/8(35x^4-30x^2+3)
(6)
P_5(x)=1/8(63x^5-70x^3+15x)
(7)
P_6(x)=1/(16)(231x^6-315x^4+105x^2-5).
(8)

When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1, 1, -1, 3, -3, 5, 3, -30, ... (OEIS A008316). The leading denominators are 1, 1, 2, 2, 8, 8, 16, 16, 128, 128, 256, 256, ... (OEIS A060818).

The first few powers in terms of Legendre polynomials are

x=P_1(x)
(9)
x^2=1/3[P_0(x)+2P_2(x)]
(10)
x^3=1/5[3P_1(x)+2P_3(x)]
(11)
x^4=1/(35)[7P_0(x)+20P_2(x)+8P_4(x)]
(12)
x^5=1/(63)[27P_1(x)+28P_3(x)+8P_5(x)]
(13)
x^6=1/(231)[33P_0(x)+110P_2(x)+72P_4(x)+16P_6(x)]
(14)

(OEIS A008317 and A001790). A closed form for these is given by

x^n=sum_(l=n,n-2,...)((2l+1)n!)/(2^((n-l)/2)(1/2(n-l))!(l+n+1)!!)P_l(x)
(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 by applying Gram-Schmidt orthonormalization without normalization to the monomials 1, x, x^2, ... on the open interval (-1,1) with weighting function 1. Before normalization, this gives the monic polynomials

Q_0(x)=1
(16)
Q_1(x)=[x-(int_(-1)^1xdx)/(int_(-1)^1dx)]·1
(17)
=x
(18)
Q_2(x)=x[x-(int_(-1)^1x^3dx)/(int_(-1)^1x^2dx)]-[(int_(-1)^1x^2dx)/(int_(-1)^1dx)]·1
(19)
=x^2-1/3
(20)
Q_3(x)=[x-(int_(-1)^1x(x^2-1/3)^2dx)/(int_(-1)^1(x^2-1/3)^2dx)](x^2-1/3)-[(int_(-1)^1(x^2-1/3)^2dx)/(int_(-1)^1x^2dx)]x
(21)
=x^3-3/5x.
(22)

Normalizing by P_n(x)=Q_n(x)/Q_n(1) gives the Legendre polynomials.

The shifted Legendre polynomials P^__n(x)=P_n(2x-1) are defined on the interval [0,1]. They obey the orthogonality relationship

int_0^1P^__m(x)P^__n(x)dx=1/(2n+1)delta_(mn).
(23)

The first few are

P^__0(x)=1
(24)
P^__1(x)=2x-1
(25)
P^__2(x)=6x^2-6x+1
(26)
P^__3(x)=20x^3-30x^2+12x-1.
(27)

The Legendre polynomials are orthogonal over (-1,1) with weighting function 1 and satisfy

int_(-1)^1P_n(x)P_m(x)dx=2/(2n+1)delta_(mn),
(28)

where delta_(mn) is the Kronecker delta.

The Legendre polynomials are a special case of the Gegenbauer polynomials with alpha=1/2, a special case of the Jacobi polynomials P_n^((alpha,beta)) with alpha=beta=0, and can be written as a hypergeometric function using Murphy's formula

P_n(x)=P_n^((0,0))(x)=_2F_1(-n,n+1;1;1/2(1-x))
(29)

(Bailey 1933; 1935, p. 101; Koekoek and Swarttouw 1998).

The Rodrigues representation provides the formula

P_l(x)=1/(2^ll!)(d^l)/(dx^l)(x^2-1)^l,
(30)

which yields upon expansion

P_l(x)=1/(2^l)sum_(k=0)^(|_l/2_|)((-1)^k(2l-2k)!)/(k!(l-k)!(l-2k)!)x^(l-2k)
(31)
=1/(2^l)sum_(k=0)^(|_l/2_|)(-1)^k(l; k)(2l-2k; l)x^(l-2k),
(32)

where |_r_| is the floor function. Additional sum formulas include

P_l(x)=1/(2^l)sum_(k=0)^(l)(l; k)^2(x-1)^(l-k)(x+1)^k
(33)
=sum_(k=0)^(l)(l; k)(-l-1; k)((1-x)/2)^k
(34)

(Koepf 1998, p. 1). In terms of hypergeometric functions, these can be written

P_n(x)=((x-1)/2)^n_2F_1(-n,-n;1;(x+1)/(x-1))
(35)
P_n(x)=(2n; n)(x^n)/(2^n)_2F_1(-n/2,(1-n)/2;1/2-n;x^(-2))
(36)
P_n(x)=_2F_1(-n,n+1;1;(1-x)/2)
(37)

(Koepf 1998, p. 3), with values at apparent singularities obtained by taking limits.

For sufficiently small |t|, a generating function for P_n(x) is given by

g(t,x)=(1-2xt+t^2)^(-1/2)=sum_(n=0)^inftyP_n(x)t^n.
(38)

Take partialg/partialt,

-1/2(1-2xt+t^2)^(-3/2)(-2x+2t)=sum_(n=0)^inftynP_n(x)t^(n-1).
(39)

Multiply (39) by 2t,

-t(1-2xt+t^2)^(-3/2)(-2x+2t)=sum_(n=0)^infty2nP_n(x)t^n
(40)

and add (38) and (40),

(1-2xt+t^2)^(-3/2)[(2xt-2t^2)+(1-2xt+t^2)]=sum_(n=0)^infty(2n+1)P_n(x)t^n
(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

sum_(n=0)^infty(P_n(x))/(n!)z^n=e^(xz)J_0(zsqrt(1-x^2)),
(42)

where J_0(x) is a zeroth order Bessel function of the first kind (Koepf 1998, p. 2).

The Legendre polynomials satisfy the recurrence relation

(l+1)P_(l+1)(x)-(2l+1)xP_l(x)+lP_(l-1)(x)=0
(43)

(Koepf 1998, p. 2). In addition,

(1-x^2)P_n^'(x)=-nxP_n(x)+nP_(n-1)(x)=(n+1)xP_n(x)-(n+1)P_(n+1)(x)
(44)

(correcting Hildebrand 1956, p. 324).

The contour integral representation can also be written

P_l(x)=1/(2pii)∮(1-2zx+z^2)^(-1/2)z^(-l-1)dz,
(45)

where the same branch and contour conditions apply. The Schläfli integral is

P_l(x)=((-1)^l)/(2^l)1/(2pii)∮((1-z^2)^l)/((z-x)^(l+1))dz,
(46)

where the contour is a counterclockwise circle enclosing x and l is a nonnegative integer.

Integrals over the interval [x,1] include the general formula

int_x^1P_m(t)dt=((1-x^2))/(m(m+1))(dP_m(x))/(dx)
(47)

for m!=0 (Byerly 1959, p. 172), from which the special case

int_0^1P_m(x)dx=(P_(m-1)(0)-P_(m+1)(0))/(2m+1)
(48)
={1 m=0; 0 m even !=0; (-1)^((m-1)/2)(m!!)/(m(m+1)(m-1)!!) m odd
(49)

follows (OEIS A002596 and A046161; Byerly 1959, p. 172). For the integral over a product of Legendre functions,

int_x^1P_m(t)P_n(t)dt=((1-x^2)[P_n(x)P_m^'(x)-P_m(x)P_n^'(x)])/(m(m+1)-n(n+1))
(50)

for m!=n (Byerly 1959, p. 172), which gives the special case

int_0^1P_m(x)P_n(x)dx={1/(2n+1) m=n; 0 m!=n, m,n both even or odd; f_(m,n) m even, n odd; f_(n,m) m odd, n even,
(51)

where

f_(m,n)=((-1)^((m+n+1)/2)m!n!)/(2^(m+n-1)(m-n)(m+n+1)[(1/2m)!]^2{[1/2(n-1)]!}^2)
(52)

(OEIS A078297 and A078298; Byerly 1959, p. 172). The latter is a special case of

int_0^1P_mu(x)P_nu(x)dx=(Asin(1/2pinu)cos(1/2pimu)-A^(-1)sin(1/2pimu)cos(1/2pinu))/(1/2pi(nu-mu)(mu+nu+1)),
(53)

where

A=(Gamma(1/2(mu+1))Gamma(1+1/2nu))/(Gamma(1/2(nu+1))Gamma(1+1/2mu))
(54)

and Gamma(z) is a gamma function (Gradshteyn and Ryzhik 2000, p. 762, eqn. 7.113.1)

Integrals over [-1,1] with weighting functions x and x^2 are given by

int_(-1)^1xP_L(x)P_N(x)dx={(2(L+1))/((2L+1)(2L+3)) for N=L+1; (2L)/((2L-1)(2L+1)) for N=L-1; 0 otherwise
(55)
int_(-1)^1x^2P_L(x)P_N(x)dx={(2(L+1)(L+2))/((2L+1)(2L+3)(2L+5)) for N=L+2; (2(2L^2+2L-1))/((2L-1)(2L+1)(2L+3)) for N=L; (2L(L-1))/((2L-3)(2L-1)(2L+1)) for N=L-2; 0 otherwise
(56)

(Arfken 1985, p. 700).

For Re[s]>0, the Laplace transform follows by integrating the finite polynomial expansion term by term,

L[P_n(t)](s)=1/(2^n)sum_(k=0)^(|_n/2_|)((-1)^k(2n-2k)!)/(k!(n-k)!s^(n-2k+1)).
(57)

A sum identity is given by

1-[P_n(x)]^2=sum_(nu=1)^n(1-x^2)/(1-x_nu^2)[(P_n(x))/(P_n^'(x_nu)(x-x_nu))]^2,
(58)

where x_nu is the nuth root of P_n(x) (Szegö 1975, p. 348). A similar identity is

sum_(nu=1)^n(1-x_nu^2)/((n+1)^2[P_(n+1)(x_nu)]^2)=1,
(59)

which is responsible for the fact that the sum of weights in Legendre-Gauss quadrature is always equal to 2.

See also

Associated Legendre Polynomial, Conical Function, Laplace-Mehler Integral, Laplace's Integral, Legendre Function of the First Kind, Legendre Function of the Second Kind, Mehler-Dirichlet Integral, Spherical Harmonic, Super Catalan Number, Zonal Harmonic

Related Wolfram sites

https://functions.wolfram.com/Polynomials/LegendreP/, https://functions.wolfram.com/Polynomials/LegendreP/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" and "Orthogonal Polynomials." 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. Reprint of the 1893 edition. https://www.gutenberg.org/ebooks/29779.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, France: 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. https://doi.org/10.1098/rstl.1870.0024.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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Legendre Polynomial

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

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