91Ó°ÊÓ

*Prove that \(u(x)=P_{n}(x)\), as defined by Rodrigues' formula, satisfies the differential equation $$ \left(1-x^{2}\right) u^{\prime \prime}(x)-2 x u^{\prime}(x)+n(n+1) u(x)=0 $$

Expand \(e^{x / 3}\) in a Laguerre series; i.e., determine the coefficients \(c_{n}\) in the formula $$ e^{x / 3} \sim \sum_{n=0}^{\infty} c_{n} L_{n}(x), \quad x \geq 0 . $$ (The formula \(\int_{0}^{\infty} e^{-a t} t^{n} d t=n ! / a^{n+1}\) may come in handy.)