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Register a new userHankel determinant and orthogonal polynomials for the Gaussian weight with a jump
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We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.
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We obtain asymptotics for Toeplitz, Hankel, and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities. Details of the proofs will be presented in another publication.
Over the last decade it has become clear that discrete Painleve equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painleve equation and determining its type according to Sakais classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakais geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painleve equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painleve equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painleve-V equation. We also provide an explicit change of variables transforming this equation to the standard form.
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
Motivated by the Gaussian symplectic ensemble, Mehta and Wang evaluated the $n$ by $n$ determinant $det((a+j-i)Gamma(b+j+i))$ in 2000. When $a=0$, Ciucu and Krattenthaler computed the associated Pfaffian $Pf((j-i)Gamma(b+j+i))$ with an application to the two dimensional dimer system in 2011. Recently we have generalized the latter Pfaffian formula with a $q$-analogue by replacing the Gamma function by the moment sequence of the little $q$-Jacobi polynomials. On the other hand, Nishizawa has found a $q$-analogue of the Mehta--Wang formula. Our purpose is to generalize both the Mehta-Wang and Nishizawa formulae by using the moment sequence of the little $q$-Jacobi polynomials. It turns out that the corresponding determinant can be evaluated explicitly in terms of the Askey-Wilson polynomials.
We consider the unital associative algebra $mathcal{A}$ with two generators $mathcal{X}$, $mathcal{Z}$ obeying the defining relation $[mathcal{Z},mathcal{X}]=mathcal{Z}^2+Delta$. We construct irreducible tridiagonal representations of $mathcal{A}$. Depending on the value of the parameter $Delta$, these representations are associated to the Jacobi matrices of the para-Krawtchouk, continuous Hahn, Hahn or Jacobi polynomials.
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