Recurrence Relations for the Legendre Polynomials

Recurrence Relations for the Legendre Polynomials. If wc differentiate both sides of equation (13.2) with respect to k we Slave 

Tills relation has been proved to hold for ft I but since the left hand side is a polynomial in ft, it must hold for all values of ft. 

If now we differentiate equation (14.2) with respect to ft wc obtain the relation 

The differentiations with respect t n h uml fi under the summation sign is justified by the fact that the series on the right-hand side of equation (13.2) is uniformly convergent for all real or complex values of h and ft which satisfy the relation [ ft 1, j h -y/ 2 — 1. 

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

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 make use of the following recurrence relations for the Legendre polynomials, in order to simplify the left hand side these are [37]... 

The reader may wish to derive the recurrence relations for the Legendre, Laguerre and Hermite polynomials from the Rodrigues expressions (6.4.4), (6.5.1) and (6.6.29). ... 

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 nodes, [qj, and weights, [wj, for the Lobatto integration rule are found from an eigenvalue problem derived from the recurrence relation for the associated Legendre polynomials, in the normalized form. 

We begin in Section 6.4.1 by reviewing, without proofs or derivations, the standard properties of the spherical harmonics and their relationship to the associated Legendre polynomials. The closely related solid harmonics are next introduced in Section 6.4.2. In Sections 6.4.3 and 6.4.4, we derive explicit Cartesian expressions for the complex and real solid harmonics, respectively. Finally, in Section 6.4.5, we derive a set of recurrence relations for the real solid harmonics. 

The expression of the solution for a given state implies only the associated Legendre polynomials PJH1 1. The use of, PJ 1 is avoided. Furthermore, a much more, simpler way of calculation, at least for the principal states Sl/2, P1/2, P3/2, -D3/2, based on a relation of recurrence, is proposed. 

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