I am trying to find values for $E$ for which I obtain polynomial solutions for the differential equation
$$(\alpha + x - \beta(1+x^2))g'(x)+(1+x^2)g''(x)+Eg(x)=0$$
I assume to have solutions of the form
$$g(x)=x^{\gamma_j}\sum_{n=0}^\infty a_{n,\gamma_j} x^n$$ where $j={1, 2}$ and $\gamma_1 =1, \gamma_2=0$. I have first tried to look at the case $j=1$ and by differentiating $g(x)$ and plug it in I get (omit $\gamma_j$ index) $$\frac{\beta-\alpha}{2}a_{0}=a_1$$ and for $n>0$ $$ (\alpha - \beta)a_n (n+1) + a_{n+1} (n+1)(n+2) + a_{n-1} (E + n^2) - \beta a_{n-2} (n-1)=0$$ at this point I am stuck...I dont know how to proceed to find out the value for $E$ for which the recurrence relation terminates. I would already be satisfied to find the value for the first few $n={1, 2, 3}$. Do I need to take also the second solution in consideration? What am I missing? Any help is deeply appreciated.