Polynomials, associated Legendre

The functions P " are associated Legendre polynomials of order m and degree I, and are associated Laguerre polynomials of degree (v — l)/2 in... 

The functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

Relationship of spherical harmonics to associated Legendre polynomials... 

Equation (E.13) relates the associated Legendre polynomial Pfip) to the (/ -I- w)th-order derivative in equation (5.58)... 

The associated Legendre polynomials P l ip) are defined in terms of the Legendre polynomials Piip) by... 

The generating functions g " p, s) for the associated Legendre polynomials may be found from equation (E.l) by letting... 

An alternative definition, but equally useful, of the associated Legendre polynomials is of die form... 

When the associated Legendre polynomials are normalized they are written in the form... 

The explicit form of the normalized associated Legendre polynomials is given by... 

The associated Legendre polynomials can be defined by the generating function... 

For m an integer the P/"(x) are polynomials. The polynomials P,°(x) are identical with the Legendre polynomials. The first few associated Legendre polynomials for x = cos 9 (a common form of Legendre s equation) are ... 

Legendre polynomials are one specific variety of a more extended class of orthogonal polynomials called associated Legendre polynomials. An associated Legendre polynomial P,m(x) is defined relative to an ordinary Legendre polynomial P,(x) through... 

The numerical generation of associated Legendre polynomials is discussed by Press et al. (1986). These authors use the following recurrence on /... 

The normalization property of associated Legendre polynomials stated above, guarantees that these functions are orthogonal over the surface of a sphere... 

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