Asymptotics of Recurrence Relation Coefficients, Hankel Determinant Ratios, and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights
Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2}, >...c,z^{-k},z^{k},...c \rbrace$ with respect to the real inner product $(f,g) \mapsto \int_{\mathbb{R}}f(s)g(s) \exp (-\mathscr{N} V(s)) \md s$, $\mathscr{N} \in \mathbb{N}$, where $V$ is real analytic on $\mathbb{R} \setminus \{0\}$, $\lim_{| x | \to \infty}(V(x)/ \ln (x^{2}
+ 1)) = +\infty$, and $\lim_{| x | \to 0} (V(x)/\ln (x^{-2}
+ 1)) = +\infty$, yields the even degree and odd degree orthonormal Laurent polynomials (OLPs), $ϕ_{2n}(z) = \sum_{k=-n}^{n} ξ^{(2n)}_{k}z^{k}$, with $ξ^{(2n)}_{n} > 0$, and $ϕ_{2n+1}(z) =
\sum_{k=-n-1}^{n} ξ^{(2n+1)}_{k}z^{k}$, with $ξ^{(2n+1)}_{-n-1} > 0$, respectively. Associated with the even degree and odd degree OLPs are two pairs of three- and five-term recurrence relations. Asymptotics in the double-scaling limit as $\mathscr{N},n \to \infty$ such that $\mathscr{N}/n = 1 + o(1)$ of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios, and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on $\mathbb{R}$, and then extracting the large-N behaviours by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].
We show that Laurent biorthogonal polynomials whose defining three-term recurrence have constant coefficients have coefficient arrays that are Riordan arrays. For each such family of Laurent biorthogonal polynomials we associate in a natural way a family of orthogonal polynomials. We also extend these results to a notion of generalized orthogonal polynomials whose recurrence depends on three parameters.
Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding ch…
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to obtain the large $n$ asymptotics of $β_n(t)$ and $\mathrm{p}(n,t)$ based on the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In addition, we show the second-order differential equation satisfied by the orthogonal…
Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent $q$-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laure…
In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^αe^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $α>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier pa…
Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall-Jacobi polynomials and their biorthogonal analogs.
In this paper we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=e^{iωx}$ on the interval $[-1,1]$, where $ω>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $ω>0$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $ω$ (called kissing points). We obtain detailed asymptotic information as $ω\to\infty$, using recent theory of multivariate highly oscillatory in…
Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered…
In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $τ$-function of the Painlevé III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients…
Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants
School of Mathematics and Statistics, Hunan University of Science and Technology（湖南科技大学数学与统计学院）；School of Mathematical Sciences, Huaqiao University（华侨大学数学科学学院）
本文研究广义Freud权对应的正交多项式与Hankel行列式，揭示离散Painlevé I hierarchy的统一结构，推导相关微分方程与恒等式，得到这些量的完全渐近展开并以十次Freud权为例说明。
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-α)(x-β)\right]^{-\frac{1}{2}}$, $x\in[0,α]$, $0<α<β$. A related system was studied by C. J. Rees in 1945, associated with the weight $\left[(1-x^2)(1-k^2x^2)\right]^{-\frac{1}{2}}$, $x\in[-1,1]$, $k^2\in(0,1)$. These are also known as elliptic orthogonal polynomials, since the moments of the weights maybe expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter $k^2$, where $0<k^2<…
We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.
It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behavior of the zeros of orthogonal polynomials and the …
In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;α>0,\;\;t>0. $$ By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of $β_n(t)$ can be expressed in terms of a particular Painlevé V transcendent. The large $n$ asymptotic expansi…
In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $α_n$ and $β_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a partic…
The coefficient of x^{-1} of a formal Laurent series f(x) is called the formal residue of f(x). Many combinatorial numbers can be represented by the formal residues of hypergeometric terms. With these representations and the extended Zeilberger's algorithm, we generate recurrence relations for summations involving combinatorial sequences such as Stirling numbers. As examples, we give computer proofs of several known identities and derive some new identities. The applicability of this method is also studied.
We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally, asymptotics for the quadrature weights are presented.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretizat…
The bivariate series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $θ(q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in $q$ of the zeros of $θ$. The coefficients $r_k$ of the stabilized series are positive integers. They are the elements of a known increasing sequence satisfying the recurrence relation $r_k=\sum _{ν=1}^{\infty}(-1)^{ν-1}(2ν+1)r_{k-ν(ν+1)/2}$.
The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from t…
In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight $$ w(x)=w(x,s):=x^{\al}\rme^{-x}\rme^{-s/x}, \quad 0\leq x<\infty, \al>0, s>0, $$ namely, the determination of the associated Hankel determinant and recurrence coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\al}\:\rme^{-x}$ 'perturbed' by a multiplicative factor $\rme^{-s/x},$ which induces an in…
We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For $t>0,$ the factor $\rex^{-t/x}$ induces an infinitely strong zero at $x=0.$ With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a syst…
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 …
We analyze $2\times 2$ Hankel-like determinants $D_n$ that arise in the initial values problem for the ultimate time survival probability $φ(u)$ in a homogeneous discrete time risk model $W(n)=u+κn+\sum_{i=1}^nZ_i$, where $Z_i$ are positive integer valued i.i.d. random claims, the initial surplus $u \in \mathbb{N}_0$ and the income rate $κ=2$. We prove the asymptotic version of a recent conjecture on the non--vanishing and monotonicity of $D_n$ and derive explicit formulas for the initial values $φ(0)$, $φ(1)$ of a recurrence that yields survival probabilities. In cases when $Z_i$ are Bernoull…
This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number o…
Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to the measure $μ$. Let $T$ be a linear operator acting in the linear space of polynomials $\PP$ and satisfying that $\dgr(T(p))=\dgr(p)-1$, for all polynomial $p$. We then construct a sequence of polynomials $(s_n)_n$, depending on $T$ but not on $μ$, such that the Wronskian type $n\times n$ determinant $\det \left(T^{i-1}(p_{m+j-1}(x))\right)_{i,j=1}^n$ is equal to the $m\times m$ determinant $\det \left(q^{j-1}_{n+i-1}(x)\right)_{i,j=1}^m$, up to multiplicative constants, where the polynomials $q_n^i$, $n,i\ge 0$, are define…
The bivariate series $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $θ(q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in $q$ of the zeros of $θ$. These series are of the form $-q^{-j}+(-1)^jq^{j(j-1)/2}(1+\sum _{k=1}^{\infty}g_{j,k}q^k)$. The coefficients of the stabilized series are expressed by the positive integers $r_k$ giving the number of partitions into parts of three different kinds. They satisfy the recurrence relation $r_k=\sum _{ν=1}^{\infty}(-1)^{ν-1}(2ν+1)r_…
