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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 ${\left({P}_{n}\right)}_{n}$ and ${\left({Q}_{n}\right)}_{n}$ whose derivatives of higher orders $m$ and $k$ (resp.) are connected by a linear algebraic structure relation such as

$\sum _{i=0}^{N}{r}_{i,n}{P}_{n-i-m}^{\left(m\right)}\left(x\right)=\sum _{i=0}^{M}{s}_{i,n}{Q}_{n-i+k}^{\left(k\right)}\left(x\right)$

for all $n=0,1,2,\cdots$, 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 ${\left({P}_{n}\right)}_{n}$ and ${\left({Q}_{n}\right)}_{n}$ (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

${D}^{k-m}\left({{\Phi }}_{M+m+i}u\right)={{\Psi }}_{N+k+i}v\left(i=0,1\right),$

the $\left(k-m\right)$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.

##### MSC:
 42C05 General theory of orthogonal functions and polynomials 33C45 Orthogonal polynomials and functions of hypergeometric type