On linearly related sequences of derivatives of orthogonal polynomials
An inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families \((P_n)_n\) and \((Q_n)_n\) whose derivatives of higher orders \(m\) and \(k\) (resp.) are connected by a linear algebraic structure relation such as \[ \sum^N_{i=0}r_{i,n}P_{n-i-m}^{(m)}(x)=\sum^M_{i=0} s_{i,n} Q^{(k)}_{n-i+k}(x) \] for all \(n=0,1,2,\dots\), where \(M\) and \(N\) are fixed nonnegative integer numbers, and \(r_{i,n}\) and \(s_{1,n}\) are given complex parameters satisfying some natural conditions. Let \(u\) and \(v\) be the moment regular functionals associated with \((P_n)_n\) and \((Q_n)_n\) (resp.). Assuming \(0\leq m\leq k\), we prove the existence of four polynomials \(\Phi_{M+m-i}\) and \(\Psi_{N+k+1}\), of degrees \(M+m+1\) and \(N+k+i\) (resp.), such that \[ D^{k-m}(\Phi_{M+m+i}u) =\Psi_{N+k+i}v(i=0,1), \] the \((k-m)\)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If \(k-m\), then \(u\) and \(v\) are connected by a rational modification. If \(k=m+1\), then both \(u\) and \(v\) are semiclassical linear functionals, which are also connected by a rational modification. When \(k+m\), the Stieltjes transform associated with \(u\) satisfies a non-homogeneous linear ordinary differential equation of order \(k-m\) with polynomial coefficients.
scientific article; zbMATH DE number 5344327
scientific article