The inverse problem for linearly related orthogonal polynomials: General case

We study the inverse problem in the theory of (standard) orthogonal polynomials involving two polynomials families $(P_n{)}_n$ and $(Q_n{)}_n$ which are connected by a linear algebraic structure such as $$P_n(x)+\sum _{i=1}^N{r}_{i,n}{P}_{n-i}(x)=Q_n(x)+\sum _{i=1}^M{s}_{i,n}{Q}_{n-i}(x)$$ for all $n=0,1, \dots$where$N$and$M$ are arbitrary nonnegative integer numbers.