zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

i=0 N r i,n P n-i-m (m) (x)= i=0 M s i,n Q n-i+k (k) (x)

for all n=0,1,2,, 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 0mk, we prove the existence of four polynomials Φ M+m-i and Ψ N+k+1 , of degrees M+m+1 and N+k+i (resp.), such that

D k-m (Φ M+m+i u)=Ψ 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:
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type