The Legendre Functions of the First Kind are solutions to the Legendre Differential Equation. If is an Integer, they are Polynomials. They are a special case of the Ultraspherical Functions with . The Legendre polynomials are illustrated above for and , 2, ..., 5.

The Rodrigues Formula provides the Generating Function

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

The Legendre polynomials are orthogonal over with Weighting Function 1
and satisfy

(9) |

A Complex Generating Function is

(10) |

(11) |

Additional integrals (Byerly 1959, p. 172) include

(12) |

(13) |

(14) |

The first few Legendre polynomials are

The first few Powers in terms of Legendre polynomials are

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.

(15) | |||

(16) | |||

(17) | |||

(18) |

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

(19) |

The associated Legendre polynomials are solutions to the associated Legendre Differential Equation,
where is a Positive Integer and , ..., . They can be given in terms of the unassociated polynomials by

(20) |

where are the unassociated Legendre Polynomials. Note that some authors (e.g., Arfken 1985, p. 668) omit the Condon-Shortley Phase , while others include it (e.g., Abramowitz and Stegun 1972, Press

(21) |

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

(22) |

(23) |

(24) |

(25) |

(26) |

(27) |

(28) |

(29) |

(30) |

Written in terms , the first few become

The derivative about the origin is

(31) |

(32) |

**References**

Abramowitz, M. and Stegun, C. 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.

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. *An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.* New York: Dover, 1959.

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.

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.

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.

Szegö, G. *Orthogonal Polynomials, 4th ed.* Providence, RI: Amer. Math. Soc., 1975.

© 1996-9

1999-05-26