Lecturer: K.S. Narain , ICTP

This lesson continues with the study of the generalized **Rodrigues' polynomials **recalling some of the results of the previous lesson and continues with the study of the **differential equations **that are satisfied by these polynomials. It was shown that the fallowing equation holds 1/W ?/?x(SW ?/?x C_{n})=-?_{n}C_{n}. As an example of the validity of this equation the **Hermite polynomials **were considered, it was shown that any second degree polynomial n can be written as a linear combination of **Hermite polynomials **to order 2 (H_{0},H_{1},H_{2}).

From the previous differential equation and using the properties of the polynomials the constant an was calculated that was the only unknown in that expression. For the case of the **Hermite polynomials **the differential equation becomes ?^{2}/?x^{2} Hn -2x?/?x Hn + 2n H_{n} = 0, for the **Laguerre polynomials** it is x ?^{2}/?x^{2} L^{v}_{n} + (-x+v+1) ?/?x L^{v}_{n} + n L^{v}_{n} = 0 and for and for the **Legendre polynomials** we have (1-x^{2}) ?^{2}/?x^{2} P_{n} + (-2x) ?/?x P_{n} -(-2n-n(n-1)+n(n+1)) P_{n}_{} = 0.

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4

Using the general differential equation for the **Rodrigues' polynomials **derived in this lesson calculate the differential equation that the **Jacobi polynomials** satisfied.