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Legendre functions of the first and second kind. Legendre differential equation. Legendre polynomials. Recurrence formulas. Generating function. Associated Legendre functions

Legendre differential equation. The equation


1)        (1 - x2)y" - 2xy' + ν(ν + 1)y = 0                   ν real

Solutions of this equation are called Legendre functions of order ν.

The Legendre equation is a special case of the associated Legendre equation


Legendre polynomials. Consider the case of 1) above when ν = n, a non-negative integer i.e. the equation


2)        (1 - x2)y" - 2xy' + n(n + 1)y = 0                  n = 0, 1, 2, 3, .....

The only solutions, with continuous first derivative for all x in the interval [-1, 1], of 2) are the Legendre polynomials, P0(x), P1(x), P2(x), .... . However, if we allow y(x) to become infinite at the end-points, then the equation is also satisfied by functions Q0(x), Q1(x), Q2(x), .... called Legendre functions of the second kind. For the case when the constant ν is real, the solutions are Legendre functions Pν(x) of the first kind (the restriction is usually added that Pν(1) =1), which are finite throughout the interval.

The Legendre polynomials are given by Rodrigue’s formula


The first few Legendre polynomials are:


If we set x = cos θ, the above can be expressed in terms of multiple angles as


It is in this form that they naturally arise in problems of temperature, potential, etc. for a sphere.

Def. Generating function. A function F that, through its representation by means of an infinite series of some sort, gives rise to a certain sequence of constants or functions as coefficients in the series.

                                                                        James & James. Mathematics Dictionary

Generating function for Legendre polynomials



Recurrence formulas for Legendre polynomials 


Orthogonality of Legendre polynomials


Because of 9), Pm(x) and Pn(x) are called orthogonal in the interval -1 ole7.gif x ole8.gif 1.

Orthogonal series of Legendre polynomials

            f(x) = A0P0(x) + A1P1(x) + A2P2(x) + ..........



Some formulas involving Legendre polynomials


where C is a simple closed curve having x as interior point.

General solution of Legendre’s equation. The general solution of Legendre’s equation is




These series converge for -1 < x < 1.

If ν is a non-negative integer ν = n = 0, 1, 2, ...... one of the series 20) or 21) terminates. In such cases,




The non-terminating series with a suitable multiplicative constant is denoted by Qn(x) and is called Legendre’s function of the second kind of order n.

Legendre function of the second kind of order n. Legendre’s function of the second kind of order n is defined for non-negative integral values of ν = n as


First several Legendre functions of the second kind


The functions Qn(x) satisfy recurrence formulas exactly analogous to 4) - 8).



Associated Legendre functions

Legendre’s associated differential equation. Legendre’s associated differential equation is


Solutions of this equation are called associated Legendre functions. We will restrict ourselves to the important case where m and n are non-negative integers.

Associated Legendre functions of the first kind. The associated Legendre functions of the first kind are given by


where Pn(x) are Legendre polynomials.

The following hold:


First several associated Legendre functions of the first kind


Generating function for ole21.gif


Recurrence formulas


Orthogonality of ole24.gif


Orthogonal series




Associated Legendre functions of the second kind. The functions


where Qn(x) are Legendre functions of the second kind.

These functions are unbounded at x = ole29.gif 1, whereas ole30.gif are bounded at x = ole31.gif 1.

The functions ole32.gif satisfy the same recurrence relations as ole33.gif [as given in 37) and 38) above].

General solution of Legendre’s associated equation



1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

2. International Dictionary of Applied Mathematics

3. Wylie. Advanced Engineering Mathematics

4. James / James. Mathematics Dictionary

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