Related papers: A Riemann-Hilbert Approach to the Akhiezer Polynom…
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-\alpha)(x-\beta)\right]^{-\frac{1}{2}}$, $x\in[0,\alpha]$, $0<\alpha<\beta$. A related system was studied by C. J. Rees in 1945,…
Sketch of proof of a theorem relating the two subjects of the title. It can be thought as an extension of results of Landau for the classical hypergeometric function. It relies on the characterization of algebraic hypergeometric functions…
We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics.
The algebraic method enables one to study the properties of the spectrum of a quadratic Hamiltonian through the mathematical properties of a matrix representation called regular or adjoint. This matrix exhibits exceptional points where it…
We consider orthogonal polynomials with respect to the weight $|z^2+a^2|^{cN}e^{-N|z|^2}$ in the whole complex plane. We obtain strong asymptotics and the limiting normalized zero counting measure (mother body) of the orthogonal polynomials…
In a previous paper [KT] we introduced determinant of the Riemann operator on Quillen's higher $K$-groups of the integer ring of an algebraic number field $K$. We showed that the determinant expresses essentially the inverse of the so…
We consider the problem of deterministically factoring a univariate polynomial over a finite field under the assumption of the Extended Riemann Hypothesis (ERH). This work builds upon the line of approach first explored by Gao in $2001$.…
In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$ (1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0, $$ with…
Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for…
A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced…
In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the…
The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to…
In this article, the matrix elements of a representation of the 5-dimensional Lie algebra K5 are obtained for the first time. The bivariate degenerate Hermite polynomials Hm(z1, z2|{\tau} ) are considered within the context of this…
We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit…
We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for…
The Fej\'{e}r-Riesz spectral factorization lemma, which represents a nonnegative trigonometric polynomial as the squared modulus of a trigonometric polynomial, was extended by Ahiezer to factorize certain entire functions and by Helson and…
A solution is given to the following problem: how to compute the multiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the…
We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials.…
This is an English translation of the paper in which N. I. Akhiezer discovered his famous orthogonal polynomials on two intervals in a connection with a generalization of the Korkin-Zolotarev (Korkine-Zolotaref) problem (see the small…
We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are…