Use Rodrigues’s formula (52) to obtain the representation

Chapter 8, Problem 34E

(choose chapter or problem)

Use Rodrigues’s formula (52) to obtain the representation (43) for the Legendre polynomials \(P_{n}(x)\). Hint: From the binomial formula,

        \(P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left\{\left(x^{2}-1\right)^{n}\right\}\)

\(=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left\{\sum_{m=0}^{n} \frac{n !(-1)^{m}}{(n-m) ! m !} x^{2 n-2 m\}}\right.\)

Equation Transcription:

Text Transcription:

Pn(x)

P_n(x)=\frac1 2^n n ! \fracd^n d x^n{(x^2-1\)^n}

=1 2^n n ! \fracd^n d x^n\left\\sum_m=0^n \fracn !(-1)^m (n-m) ! m ! x^2 n-2 m\}