Chapter 12.22, Problem 24P

Show that the functions $L_n^k(x)$ are orthogonal on $(0,\infty )$ with respect to the weight function $x^k{e}^{-x}$.

Hint: Write the differential equation (22.26) as $x^{-k}{e}^x\frac{d}{dx}\left(x^{k+1}{e}^{-x}{y}^{\prime }\right)+ny=0$ and see Sections 7 and 19.