Legendre polynomials orthogonality property

The powers of the variable x (1, x, x2,..., xa,...) are not orthogonal functions over a unique interval. However, particular sets of polynomials present the orthogonality property. A simple and useful example is that of Legendre polynomials. Let us choose over the range [—1, +1] the first two polynomials... 

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

Here, A / are the order parameters, and Piicos Op) are the Legendre polynomials. Using orthogonal polynomials properties and the hinction in Equation 12.7 one can obtain the stationary order parameters in the AHB approximation ... 

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,... 

Substitution of this into Eq. (3.70) and using the orthogonality properties of Legendre polynomials given in Eq. (3.13), we obtain the infinite set of differential-difference equations... 

The coefficients ij, are obtained in the usual way by applying the orthogonality properties of the Legendre polynomials. It follows that... 

There are a number of different kinds of orthogonal polynomials one can use, including continuous polynomials like Lagrange or Legendre polynomials [22], [21], and discrete ones, such as Hahn s polynomial [23]. The orthogonality property allows one to obtain the roots of the polynomial Xi, i =, 2,..., m — 1. Since orthogonal polynomials are also formed by linear combination of x or (for simplicity we can take the example of polynomials in x), Equation 2.30 can be rewritten at each collocation point in terms of new coefficients di... 

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