Legendre functions, associated orthogonality

Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

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

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,... 

Use of the orthogonality properties of the associated Legendre functions, which are... 

Fm basis sets corresponding to classical orthogonal pol3momials (harmonic oscillator functions, Qiebyshev polynomi, Legendre and associated L endre lynomials, etc.) the approximation 2.5 is equivalent to the approximatimi of V by Gaussian quadrature. For... 

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 ... 

The orthogonality property of the Wigner d-functions is similar to that of the associated Legendre functions and is given by... 

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

Simulations have also been used to examine the local dynamics of individual ions in the liquid state. In an early study from Lynden-Bell s group [12], rotational dynamics of the cations and anions of [EMIM][d] and [MMIM][PF6] were examined by computing averages of the Legendre polynomials Pi (cos 9 t)) and Pi (cos P (t)) as a function of time, where 9 (t) is the angle moved through by one of three orthogonal vectors associated with an ion. The decorrelation of these order parameters was fitted to an exponential, and rotational relaxation times ti and ti were determined. At... 

These representations would all seem to be equivalent each is associated with certain advantages. The system of Euler s angles has a long history and forms the basis of the well-known set of orthogonal functions, the associated Legendre polynomials. Mathematical representation of orientation distributions are... 

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