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Register a new userA Simple Crystalline Measure
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Added by
Alexander Ulanovskii
Publication date
2020
fields
Physics
and research's language is
English
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We prove that every pair of exponential polynomials with imaginary frequencies generates a Poisson-type formula.
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We prove a sharp Lieb-Thirring type inequality for Jacobi matrices, thereby settling a conjecture of Hundertmark and Simon. An interesting feature of the proof is that it employs a technique originally used by Hundertmark-Laptev-Weidl concerning sums of singular values for compact operators.
A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equation, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions to these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series $_4F_3$ at 1. Keywords: Fuchsian differential equation, hypergeometric differential equation, middle convolution, Dotsenko-Fateev equation, recurrence formula, series solution
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
The principal aim of this paper is to extend Birmans sequence of integral inequalities originally obtained in 1961, and containing Hardys and Rellichs inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand side of the inequality as well. Our new technique of proof for this sequence of inequalities relies on a combination of transforms originally due to Hartman and Muller-Pfeiffer. The results obtained considerably improve on prior results in the literature.
An algebra denoted $mmathfrak{H}$ with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on $mmathfrak{H}$. Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of $mmathfrak{H}$ in $mathcal{U}(mathfrak{sl}_2)$ is presented. A Pade approximation table for the binomial function is obtained as a by-product.
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