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.