Legendre functions orthogonality relation

Substituting Eq. (267) into Eq. (265), taking the inner product, and utilizing the orthogonal properties and known recurrence relations [51] for the associated Legendre functions Pf cosi ) and the Hermite polynomials H (z) then yields the infinite hierarchy of differential recurrence relations for the clnm(t) governing the orientational relaxation of the system, namely,... 

The associated Legendre functions satisfy an orthogonality relation ... 

In the preceding sections we introduced several integrals and orthogonality relations of associated Legendre functions, which we shall prove in the following ... 

Using the plane wave expansion (C.l), the orthogonality relations of the associated Legendre functions and the equation Ke roi = K zoi, we derive... 

In recent years, the first order finite difference methods have been superseded by nodal methods. In these the flux in each mesh element, or node, is represented by a set of orthogonal functions, such as Legendre polynomials for each direction, or other types of expansion. Using such flux representations, a more accurate solution can be obtained using a coarser mesh. The matrix equations relating the various components of the flux become more complicated, involving a relationship between components of the flux inside the node and on the surfaces. 

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