Chapter 8.8, Problem 11E

Show that the confluent hypergeometric equation

$x{y}^{″}+\left(\gamma -x\right)y^{\prime }-\alpha y=0$,

where $\alpha$ and $\gamma$ are fixed parameters and $\gamma$ is not an integer, has two linearly independent solutions

$y_1\left(x\right)=F_1\left(\alpha ;\gamma ;x\right):=1+{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{{\left(\alpha \right)}_n}{n!{\left(\gamma \right)}_n}{x}^n}$

and

$y_2\left(x\right)=x_1^{1-\gamma }{F}_1\left(\alpha +1-\gamma ;2-\gamma ;x\right)$.