I read a book where there is this example:
The recurrence relation holds for polynomials $\pi_1,\pi_2,\dots$ with $\deg(\pi_n)=n$
$$x\pi_n(x)=a_n\pi_{n+1}(x)+b_n\pi_n(x)+a_{n-1}\pi_{n-1}(x)$$
then these are orthonormal polynomials with $a_n=\frac{\gamma_n}{\gamma_{n+1}}$ where $\pi_n(x)=\gamma_nx^n+\dots$.
I have especially troubles to understand why $a_n=\frac{\gamma_n}{\gamma_{n+1}}$ and there is no derivation.
Edit: I remove my solution since it is an exercise. I keep the final result though.
I can show the recursion for monic polynomials $p_n(x)$.
$$xp_n(x)=a_np_{n+1}(x)+b_np_n(x)+c_{n}p_{n-1}(x)$$
But here I am lost.
Hint $$ x\pi_n(x)=a_n\pi_{n+1}(x)+b_n\pi_n(x)+a_{n-1}\pi_{n-1}(x) $$ What is the coefficient of $x^{n+1}$ on the LHS? What about on the RHS?
Make them equal and then you are done.