Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals ($|q|=1$) are considered.
We present a new explicit family of polynomials orthogonal on the unit circle with a dense point spectrum. This family is expressed in terms of q-hypergeometric function of type ${_2}ϕ_1$. The orthogonality measure is the wrapped geometric distribution. Some "classical" properties of the above polynomials are presented.
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a new type of elliptic hypergeometric function. We show that obtained polynomials are orthogonal on the unit circle with respect to a dense point meausure, i.e. the spectrum consists from infinite number points of increase which are dense on the unit circle. We construct also corresponding explicit systems of polynomials orthogonal on the interval of the real ax…
It is stated and proved a characterization theorem for Laguerre-Hahn orthogonal polynomials on non-uniform lattices. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.
In this paper, we study a family of orthogonal polynomials $\{ϕ_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $ϕ_n(z)$ as the polynomial degree $n$ tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomia…
In this paper, we develop the Riemann-Hilbert method to study the asymptotics of discrete orthogonal polynomials on infinite nodes with an accumulation point. To illustrate our method, we consider the Tricomi-Carlitz polynomials $f_n^{(α)}(z)$ where $α$ is a positive parameter. Uniform Plancherel-Rotach type asymptotic formulas are obtained in the entire complex plane including a neighborhood of the origin, and our results agree with the ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and [{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].
This is the full version of "Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results" appearing on this server and also published in IMRN 2003, no. 15, pp. 821-858.
We obtain the sharp lower bound for the uniform norm of the orthogonal polynomials in the Steklov class. We also prove the sharp estimates for the polynomial entropy in this class.
We study solutions of three-term recurrence relations whose $N$-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more. Namely, we find the constructive formula for the density of their orthogonality measure in terms of Turán determinants and we determine their exact asymptotic behavior. We treat both bounded and unbounded cases in a uniform manner.
Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision, roughly speaking, $O(k^{-2/3}).$
In the case when the weight and its inverse belong to BMO(T), we prove the asymptotics of the monic orthogonal polynomials in L^p, 2<p<p_0. Immediate applications include the estimates on the uniform norm and asymptotics for the polynomial entropy.
We study a new family of q-Meixner multiple orthogonal polynomials of the first kind. The discrete orthogonality conditions are considered over a non-uniform lattice with respect to different q-analogues of Pascal distributions. We address some algebraic properties, namely raising and lowering operators as well as Rodrigues-type. Based on the explicit expressions for the raising and lowering operator a high-order linear q-difference equation with polynomial coefficients for the q-Meixner multiple orthogonal polynomials of the first kind is obtained. Finally, we obtain the nearest neighbor recu…
We prove that if $E$ is a planar self-similar set with similarity dimension $d$ whose defining maps generate a dense set of rotations, then the $d$-dimensional Hausdorff measure of the orthogonal projection of $E$ onto any line is zero. We also prove that the radial projection of $E$ centered at any point in the plane also has zero $d$-dimensional Hausdorff measure. Then we consider a special subclass of these sets and give an upper bound for the Favard length of $E(ρ)$ where $E(ρ)$ denotes the $ρ$-neighborhood of the set $E$.
We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite unio…
We investigate the uniform asymptotic of some Sobolev orthogonal polynomials. Three term recurrence relation is given, moreover we give a recurrence relation between the so-called Sobolev orthogonal polynomials and Freud orthogonal polynomials.
A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidate…
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrodinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.
A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{\cal A}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lin…
The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christoffel-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large $n$ limit. These random walks are beyond b…
We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic expansion for these polynomials, valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its and Kitaev.
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.
Let $p_n(x)$ be orthogonal polynomials associated to a measure $dμ$ of compact support in $R$. If $E\not\in supp(dμ)$, we show there is a $δ>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-δ, E+δ)$. If $E$ is an isolated point of $supp(dμ)$, we show there is a $δ$ so that for all $n$, either $p_n$ or $p_{n+1}$ has at most one zero in $(E-δ, E+δ)$. We provide an example where the zeros of $p_n$ are dense in a gap of $supp(dμ)$.
