## 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 $$0\leq m\leq k$$, 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.

### MSC:

 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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### References:

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