$$ x p_n(x) = a_n p_n+1(x) + b_n p_n(x) + a_n-1 p_n-1(x). $$
The classical moment problem is solved in principle, but many related questions remain active research topics: $$ x p_n(x) = a_n p_n+1(x) + b_n p_n(x) + a_n-1 p_n-1(x)
The asks: Given a sequence of real numbers $(m_n)_n=0^\infty$, does there exist a positive measure $\mu$ on $\mathbbR$ (or a subset thereof) such that: \quad z \in \mathbbC\setminus\mathbbR $$
Why? Because for any polynomial $P(x) = \sum_k=0^n a_k x^k$, we have: $$ x p_n(x) = a_n p_n+1(x) + b_n p_n(x) + a_n-1 p_n-1(x)
$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$