Rodrigues’ formula

The Jacobi polynomials are given explicitly by the Rodrigues formula 

The curves for these three Jacobi polynomials are shown in Fig. 8.3 over the domain [0,1]. It is important to note that /j has one zero, has two zeros, and has three zeros within the domain [0,1] zeros are the values of x, which cause Jf (x) = 0. These zeros will be used later as the interior collocation points for the orthogonal collocation method. 

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Rodrigues formula for the Legendre polynomials may be derived as follows. Consider the expression... 

Rodrigues formula is of great use in the evaluation of definite integrals involving Legendre polynomials. Consider, for instance, the integral... 

From Rodrigues formula (15.3) we derive the simple expression... 

There are many ways to prove this proposition. Our proof is straightforward, elementary and rather ugly. For a more elegant proof via the Rodrigues formula, see [WW, Chapter XV] or [DyM, Section 4.12]. 

Some features of the Legendre and Laguerre polynomials are discussed next. The Rodrigues formula for associated Legendre polynomials is... 

Finally, using the Rodrigues formula for associated Laguerre polynomials (cf. Abramowitz and Stegun, 1965 Powell and Craseman, 1961)... 

With a choice of constant such that P(. ) = 1, the Legendre polynomials are defined by Rodrigues formula ... 

Together with the Rodrigues formula (see e.g. Abramowitz and Stegun [1]) for Hermite polynomials the derivatives of the Gaussian function are given by... 

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