Chapter 12.22, Problem 17P

In (22.23), write the series for the exponential and collect powers of $h$ to verify the first few terms of the series. Verify the identity $x\frac{{\partial }^2\Phi }{\partial x^2}+(1-x)\frac{\partial \Phi }{\partial x}+h\frac{\partial \Phi }{\partial h}=0.$

Substitute the series in (22.23) into this identity to show that the functions $L_n(x)$ in (22.23) satisfy Laguerre’s equation (22.21). Verify that the constant term is 1 by putting $x=0$ in the generating function. [You have then proved that the functions called $L_n(x)$ in (22.23) are really Laguerre polynomials since, by Problem 15, (22.21) has just one polynomial solution of degree n.]