We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight \[ω(x;t,λ)=x^λ\exp\left(-\tfrac13x^3+tx\right),\qquad x\in \mathbb{R}^+,\] with parameters $λ>-1$ and $t\in \mathbb{R}$. We also investigate the zeros and recurrence coefficients of the polynomials. The generalised sextic Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(-x^6+tx^2\right), \qquad x\in \mathbb{R},\] arises from a symmetrisation of the generalised Airy weight and we study analogous properties of the polynomials orthogonal with respect to this weight.
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values of Gegenbauer polynomials. Similar problem for products of Airy functions is solved in terms of terminating hypergeometric series.
Asymptotic formulas are derived for the Stieltjes-Wigert polynomials $S_n(z;q)$ in the complex plane as the degree $n$ grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the $q$-Airy function $A_q(z)$ is used as the approximant. For real $x> 1/4$, a limiting relation is also established between the $q$-Airy function $A_q(x)$ and the ordinary Airy function $\mathrm{Ai}(x)$ as $q \to 1$.
We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degenerations. Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.
Jacobi-Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.
In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of classes of functions defined by the order of best approximation by algebraic polynomials.
We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities…
A method of generating differential operators is used to solve the spectral problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we find a linear differential operator with polynomial coefficients of the first order that has a finite sequence of polynomial eigenfunctions generalising the operator considered by M. Kac. In addition, we explain spectral properties of two related tridiagonal matrices whose shape differ from our generalisation.
We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalised sextic Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(-x^6+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $λ>-1$ and $t\in\mathbb{R}$. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalised hypergeometric functions ${}_1F_2(a_1;b_1,b_2;z)$. We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlightin…
Generalised matrix elements of the irreducible representations of the quantum $SU(2)$ group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of $sl(2)$. A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. {\bf 66}, Ser. A (1990), pp. 146--149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and …
In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^α(1-x)^β, $$ where $t \ge 0$, $α>0$, $β>0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, diff…
In this paper, we compute the probability that an $N \times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem.
In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko. They then use their result to obtain a generalisation of the famous Selberg integral. In this note we point out that the Selberg-generalisation of Richards and Zheng is a special case of an integral over Jack polynomials due to Kadell. We then show how an integral formula for Jack polynomials of Okounkov and Olshanski may be applied to prove Kadell's integral along the lines of Richards and Zheng.
Given two real numbers, the $L^2$ functions whose Fourier transforms vanish with a certain rapidity near the given numbers are characterised as those that are expressible as the sum of a certain number of generalised finite differences that is independent of the function. These generalised differences can be regarded as approximating the appropriate powers of first order ordinary differential operators. The upshot of this is that for operators in a certain class of ordinary differential operators that have polynomial multipliers, their ranges on the Sobolev spaces corresponding to the operator…
We consider polynomials that are orthogonal on $[-1,1]$ with respect to a modified Jacobi weight $(1-x)^α(1+x)^βh(x)$, with $α,β>-1$ and $h$ real analytic and stricly positive on $[-1,1]$. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval $[-1,1]$, for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants. For the asymptotic analysis we use the steepest descent technique for Riemann--Hilbert problems developed by Deift and Zhou, and applied to …
