The inverse problem for linearly related orthogonal polynomials: General case

Abstract: 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.