Consider the differential equation
$$xf''(x)+(\alpha+1-x)f'(x)+nf(x)=0 ..........(*)$$,
we know that the Laguerre Polynomial $L_n^{\alpha}(x)=\frac{x^{-\alpha}e^x}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+\alpha})$ is a solution to the above differential equation. Consider the confluent hypergeometric series with parameter $(a,b)$:
$$F(a,b,x)=\sum_{k=0}^{\infty}\frac{(a)_kx^k}{(b)_kk!},$$
where $(a)_n=a(a+1)\cdots (a+n-1)$ is the Pochhammer symbol. We also know that $F(a,b,x)$ satisfies the differential equation $xf''(x)+(b-x)f'(x)-af(x)=0.....(**)$.
The question is: find another solution of $(*)$ of type $x^cF(A,B,x)$.
I have no background in differential equation, so i have no idea where to start, please helps.