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On linearly related sequences of derivatives of orthogonal polynomials. (English) Zbl 1160.42011
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 $0\le m\le k$, 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.

42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
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