In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^α e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N} \}$. We focus on the "band-saturated region" situation when the parameter $c > \frac{π^2}{4}$. As $n \to \infty$, uniform expansions for $P_{n, n}(z)$ are achieved for $z$ in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for $z$ near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient $h_{n, N}$, recurrence coefficients $\mathscr{B}_{n, N}$ and $\mathscr{A}_{n, N}^2$, are also obtained. Our method is based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems.
C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted $L^{p}$ spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre--Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán.
The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form.
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.
Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(α)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and $α$ small or large, uniformly for unbounded real and complex values of $x$. The new expansions extend the range of computability of $L_n^{(α)}(x)$ compared to previous expansions, in particular with respect to higher terms and large values of $α$. Numerical evidence of their accuracy for real and complex values of $x$ is provided.
A conjecture of I. Krasikov is proved. Several discrete analogues of classical polynomial inequalities are derived, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-Pólya class.
In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of the classical Laguerre polynomials and deduce the coefficients of their corresponding five-term recurrence relation, as well as the asymptotic behavior of these coefficients when the degree of the polynomials tends to infinity. Also, the outer relative asymptotics of or…
A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.
We look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-discrete Painlevé III equation. Next we consider the q-Laguerre or generalized Stieltjes-Wigert weight functions with a quadratic transformation and derive recursive equations for the recurrence coefficients of the orthogonal polynomials. These turn out to be related to the q-discrete Painlevé V eq…
In this paper we consider a semi-classical variation of the weight related to the little $q$-Laguerre polynomials and obtain a second order second degree discrete equation for the recurrence coefficients in the three-term recurrence relation.
Introducing the discrete probability distribution by means of the Prabhakar (or the three--parameter Mittag--Leffler) function, we establish explicit expressions for raw and factorial moments and also general fractional order moments. Applying an elementary moment inequality we obtain functional upper bounds for the Turánian difference for Prabhakar function. Finally, a Laguerre inequality is proved and functional upper bound has been given for Laguerreian difference for Prabhakar function.
It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!} \ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two consecutive zeros of $f(x)$ is at least $\sqrt{4-\frac{6}{m+2}}.$ These inequalities are a surprisingly similar discrete analogue of higher degree generalizations of the Laguerre and Turan inequalities. Being applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds uniform in all parameters involved, on the polynomials and …
We derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.
The Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometrical type. We derive the Laguerre-Freud equations when the first moments of the weights are given by the ${}_1F_2$, ${}_2F_2$ and ${}_3F_2$ generalized hypergeometrical functions.
We propose a simple uniform lower bound on the spacings between the successive zeros of the Laguerre polynomials $L_n^{(α)}$ for all $α>-1$. Our bound is sharp regarding the order of dependency on $n$ and $α$ in various ranges. In particular, we recover the orders given in \cite{ahmed} for $α\in (-1,1]$.
For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel-Lieb-Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.
This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the spe…
We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre …
In this paper the Riemann-Hilbert problem, with jump supported on a appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights ---which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann-Hilbert problem are derived. An explicit and general example is presented to illustrate the theoretical results of the work. Related matrix eigenvalue problem…
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates f…
We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure $ρ_{a,c}$ and Hahn measure $ρ_{a,b,N}$ by removing a finite number of their mass points when the parameter $c$ of the Meixner measure and $a$ or $b$ of the Hahn measure are positive integers. From the new Krall-Meixner families we construct new families of exceptional Meixner and Laguerre pol…
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…
We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme, with the exception of the Jacobi, Bessel, Laguerre, and Hermite polynomials.
The q-Laguerre polynomials correspond to an indetermined moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's ${}_1ψ_1$-summation we complement the orthogonal q-Laguerre polynomials into an explicit orthogonal basis for the corresponding L^2-space. The dual orthogonal system consists of so-called big q-Bessel functions, which can be obtained as a rigorous limit of the orthogonal system of big q-Jacobi polynomials. Interpretations on the SU(1,1) and E(2) quantum groups are discussed.
Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra $\mathfrak{su}(1,1)$. The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of $\mathfrak{su}(1,1)$ are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and fun…
Classical settings of discrete and continuous orthogonal expansions, like Laguerre, Bessel and Jacobi, are associated with second order differential operators playing the role of the Laplacian. These depend on certain parameters of type that are usually restricted to a half-line, or a product of half-lines if higher dimensions are considered. Following earlier research done by Hajmirzaahmad, we deal in this paper with Laplacians in the above-mentioned contexts with no restrictions on the type parameters and bring to attention naturally associated orthogonal systems that in fact involve the cla…
As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual stat…
The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in…
In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be …
We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polinomials, which are uniform in all the parameters.
We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2tw_n + w_n(w_n+1 + w_n + w_n-1) for the recurrence coefficients p_n+1 = xp_n - w_np_n-1 of the orthogonal polynomials related to the weight w(x) = exp(-4(tx^3 + x^4)) (notation of [26, pp. 34-36])…
The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, …
In this paper, we provide a family of generalized discrete $q$-Hermite II polynomials denoted by $\tilde{h}_{n,α}(x,y|q)$. An explicit relations connecting them with the $q$-Laguerre and Stieltjes-Wigert polynomials are obtained. Summation formula is derived by using different analytical means on their generating functions.
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $\pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of ge…
Given a finite set $F=\{f_1,\cdots ,f_k\}$ of nonnegative integers (written in increasing size) and a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant $\det(p_{f_i}(x+j-1))_{i,j=1,\cdots,k}$. In this paper we prove a nice invariant property for this kind of Casorati determinants when the set $F$ is changed by $I(F)=\{0,1,2,\cdots,\max F\}\setminus \{\max F-f:f\in F\}$. This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Chr…
The coefficients that appear in uniform asymptotic expansions for integrals are typically very complicated. In the existing literature the majority of the work only give the first two coefficients. In a limited number of papers where more coefficients are given the evaluation of the coefficients near the coalescence points is normally highly numerically unstable. In this paper, we illustrate how well-known Cauchy type integral representations can be used to compute the coefficients in a very stable and efficient manner. We discuss the cases: (i) two coalescing saddles, (ii) two saddles coalesc…
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 our previous papers, the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials were presented. These identities are naturally derived through quantum mechanical formulation of the classical orthogonal polynomials; ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the $q$…
We present 15 explicit examples of discrete time Birth and Death processes which are exactly solvable. They are related to the hypergeometric orthogonal polynomials of Askey scheme having discrete orthogonality measures. Namely, they are the Krawtchouk, three different kinds of q-Krawtchouk, (dual, q)-Hahn, (q)-Racah, Al-Salam-Carlitz II, q-Meixner, q-Charlier, dual big q-Jacobi and dual big q-Laguerre polynomials. The birth and death rates are determined by the difference equations governing the polynomials. The stationary distributions are the normalised orthogonality measures of the polynom…
The multi-indexed orthogonal polynomials (the Meixner, little $q$-Jacobi (Laguerre), ($q$-)Racah, Wilson, Askey-Wilson types) satisfying second order difference equations were constructed in discrete quantum mechanics. They are polynomials in the sinusoidal coordinates $η(x)$ ($x$ is the coordinate of quantum system) and expressed in terms of the Casorati determinants whose matrix elements are functions of $x$ at various points. By using shape invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point $x$.…
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.
As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.
As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schrödinger equation with negative `eigenvalues', ex…
