Legendre functions recurrence relations

Recurrence Relations for the Function Recurrence relations for the Legendre function of the second kind can be derived from Neumann s formula (18.1) anti the corresponding recurrence relations for the Legendre polynomials Pn(fi). From the recurrence relation (It.2) and Neumann s formula we have... 

We next derive some recurrence relations for the Legendre polynomials. Differentiation of the generating function g p, s) with respect to s gives... 

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,... 

Recurrence relations have been derived for the Cmn,mn coefficients [150]. The derivation is simple for the case of axial translation and positive m. Using the integral representation (B.60) (or (B.63)) and the recurrence relations for the normalized associated Legendre functions (A.15) and (A.16), give... 

We note that the coefficients ai(-) and fei( ) can be expressed in terms of the coefficients a( ) by making use of the recurrence relations for the associated Legendre functions. As in the scalar case, the integration with respect to the azimuthal angle a gives... 

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