Topic
Difference polynomials
About: Difference polynomials is a research topic. Over the lifetime, 7911 publications have been published within this topic receiving 120549 citations.
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01 Oct 1991TL;DR: In this article, the orthogonality relation (2.0.1) is reduced to 2.0, where w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ i at the points x = x i.
Abstract: The basic properties of the polynomials p n (x) that satisfy the orthogonality relations
$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m
e n) $$
(2.0.1)
hold also for the polynomials that satisfy the orthogonality relations of a more general form, which can be expressed in terms of Stielties integrals
$$ \int_a^b {{p_n}(x)} {p_m}(x)dw(x) = 0\quad (m
e n), $$
(2.0.2)
where w(x) is a monotonic nondecreasing function (usually called the distribution function). The orthogonality relation (2.0.2) is reduced to (2.0.1) in the case when the function w(x) has a derivative on (a, b) and w′(x) = ϱ(x). For solving many problems orthogonal polynomials are used that satisfy the orthogonality relations (2.0.2) in the case when w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ i at the points x = x i . In this case the orthogonality relation (2.0.2) can be rewritten in the form
$$ \sum\limits_i {{p_n}({x_i})pm} ({x_i}){\rho_i} = 0\quad (m
e n). $$
(2.0.3)
1,032 citations
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TL;DR: In this article, a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type was studied.
Abstract: We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi polynomials in the limit $q=-1$. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for $q=-1$.
775 citations
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01 Jun 1994
TL;DR: General concept of polynomials elementary inequalities zeros of poynomials special classes of polynnomials extremal problems for polynmials inequalities connected with trigonometric sums are introduced.
Abstract: General concept of polynomials elementary inequalities zeros of polynomials special classes of polynomials extremal problems for polynomials inequalities connected with trigonometric sums.
508 citations