Mourad E. H. Ismail

• University of Central Florida

In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality measure of these polynomials, as well as the spectral measure for the associated doubly infinite tridiagonal J...

We study the behavior of the smallest possible constants d(a,b) and d_{n} in Hardy’s inequalities \int_{a}^{b}\biggl(\frac{1}{x}\int_{a}^{x}f(t)dt\biggr)^{2}\:dx\leq d(a,b){}\int_{a}^{b} [f(x)]^{2}\: dx and \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_{j}\Big)^{2}\leq d_{n}{}\sum_{k=1}^{n}a_{k}^{2}. The exact constant d(a,b) and the precise rate o...

In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches -1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{...

We develop Taylor type series expansions for entire functions of order zero using general interpolation sequences that diverge to infinity sufficiently fast. We also derive two-point and multi-point Lidstone type series expansions for entire functions of order zero. The coefficients of these Taylor and Lidstone type series are expressed in terms of...

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^bf^2(x)dx $$ and $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2. $$ The exact constant $d(a,b)$ and the exact rate of convergence of $d...

In this paper, we expand functions of specific q-exponential growth in terms of its even (odd) Askey-Wilson q-derivatives at 0 and η=(q1/4+q−1/4)/2. This expansion is a q-version of the celebrated Lidstone expansion theorem, where we expand the function in q-analogs of Lidstone polynomials, i.e., q-Bernoulli and q-Euler polynomials as in the classi...

We use the tridiagonal representation approach to solve the radial Schr\"odinger equation for the continuum scattering states of the Kratzer potential. We do the same for a radial power-law potential with inverse-square and inverse-cube singularities. These solutions are written as infinite convergent series of Bessel functions with a discrete inde...

We introduce three one parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey–Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey–Wilson polynomials, a...

With certain constraints on the parameters μ4 and μ2, it is known that the Schrödinger equation with the sextic anharmonic oscillator potential V(r)=r6+μ4r4+μ2r2 is quasi-exactly solvable. Here, we solve the Schrödinger equation for arbitrary values of the potential parameters in the d-dimensional case. The method discussed offers a practical solut...

We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials whose moment problem is indeterminate. We provide several orthogonality measures for the infinite family and d...

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn’s continued fraction contains as special cases the famous Rogers–Ramanujan continued fraction and two of Ramanujan’s generalizations. The orthogonality measure of the set of polynomials obtained has an absolutely continuous component. We find generati...

In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $\eta=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion theorem, where we expand the function in $q$-analogs of Lidstone polynomials, i.e., q-Bernoulli and $q$-Euler poly...

In this paper, we use an identity connecting a modified [Formula: see text]-Bessel function and a [Formula: see text] function to give [Formula: see text]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [Formula: see text]-version of a partition identity. We prove new relations and identities invo...

We show that there is a concept of q-translation behind the approach used by Liu to prove summation and transformation identities for q-series.We revisit the q-translation associated with the Askey–Wilson operator introduced in Ismail (Ann Comb 5(3–4):347–362, 2001), simplify its formalism and point out new properties of this translation operator.

We introduce three one-parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey-Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey-Wilson polynomials. W...

In this article, we present a necessary and sufficient condition under which sequences are minimal completely monotonic.

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multiva...

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt’s volumes on Ramanujan’s notebooks. We give two alternate approaches to proving Ramanujan’s Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We conside...

In this paper, we employ a difference equation approach to study the Plancherel‐Rotach asymptotics of ‐orthogonal polynomials about their largest zeros. Our method for ‐difference equations is an analogue to the turning point problem for Hermite differential equations. It works well in the toy problems of Stieltjes‐Wigert polynomials and ‐Hermite p...

A functionFunctionf(x) is called completely monotonic if (−1)mf(m)(x) > 0. In random matrix theory when the associated orthogonal Orthogonal polynomials have Freud weights, it is known that the expectation of having m eigenvalues Eigenvalues of a random Hermitian Hermitian matrix in an interval is a multiple of (−1)m times the m-th derivative Deriv...

We derive the connection relations between three q-Taylor polynomial bases. We also derive the connection relations between two of these bases and the continuous q-Hermite polynomials, as well as several new generating functions for the continuous q-Hermite polynomials. Our results also lead to some new integral evaluations. Using one of these conn...

In this paper, we derive some properties of a Ramanujan type entire function. A mild generalization of the Garret-Ismail-Stanton m-version of the Rogers-Ramanujan identities is obtained. Moreover, we investigate the zeros of the Ramanujan type entire function, and our results generalize those for the zeros of the Ramanujan function. Finally, an int...

We introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a...

We introduce a finite difference and $q$-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or $q$-difference equation subject to a "terminating condition", which is precisely defined. Wh...

The Asymptotic Iteration Method (AIM) is a technique for solving analytically and approximately the linear second-order differential equation, especially the eigenvalue problems that frequently appear in theoretical and mathematical physics. The analysis and mathematical justifications of the success and failure of the asymptotic iteration method a...

The asymptotic iteration method is a technique for solving analytically and approximately the linear second-order differential equation, especially the eigenvalue problems that frequently appear in theoretical and mathematical physics. The analysis and mathematical justifications of the success and failure of the asymptotic iteration method are det...

We show that there is a concept of q-translation behind the approach used by Liu to prove summation and transformation identities for q-series. We revisit the q-translation associated with the Askey–Wilson operator introduced in Ismail (Ann Comb 5(3–4):347–362, 2001), simplify its formalism and point out new properties of this translation operator.

A general family of matrix valued Hermite type orthogonal polynomials is introduced as the matrix orthogonal polynomials with respect to a weight. The matrix polynomials are eigenfunctions of a matrix differential equation. For the weight we derive Pearson equations, which allow us to derive many explicit properties of these matrix polynomials. In...

We study a continued fraction due to Ramanujan, that he recorded as Entry 12 in Chapter 16 of his second notebook. It is presented in Part III of Berndt's volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanujan's Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We conside...

We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to b...

We study the resolvent and spectral measure of certain doubly infinite Jacobi matrices via asymptotic solutions of two-sided difference equations. By finding the minimal (or subdominant) solutions or calculating the continued fractions for the difference equations, we derive explicit formulas for the matrix entries of resolvent of doubly infinite J...

In this paper, we develop a general theory of quasi-orthogonal polynomials. We first derive three-term recurrence relation and second-order differential equations for quasi-orthogonal polynomials. We also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials. In addition, we in...

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn's continued fraction contains as special cases the famous Rogers--Ramanujan continued fraction and two of Ramanujan's generalizations. The orthogonality measure of the set of polynomials obtained has an absolutely continuous component. We find generat...

A general family of matrix valued Hermite type orthogonal polynomials is introduced and studied in detail by deriving Pearson equations for the weight and matrix valued differential equations for these matrix polynomials. This is used to derive Rodrigues formulas, explicit formulas for the squared norm and to give an explicit expression of the matr...

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also deve...

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analo...

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also deve...

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials...

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials...

Calogero and his collaborators recently observed that some hypergeometric polynomials can be factored as a product of two polynomials, one of which is factored into a product of linear terms. Chen and Ismail showed that this property prevails through all polynomials in the Askey scheme. We show that this factorization property is also shared by the...

This paper contains a brief review of orthogonal polynomials in two and several variables. It supplements the Koornwinder survey [40]. Several recently discovered systems of orthogonal polynomials have been treated in this work. We did not provide any proofs of the theorem presented here but references to the original sources are given for the bene...

We investigate some combinatorial and analytic properties of the n-dimensional Hermite polynomials introduced by Hermite in the late 19-th century. We derive combinatorial interpretations and recurrence relations for these polynomials. We also establish a new linear generating function and a Kibble–Slepian formula for the n-dimensional Hermite poly...

We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple and lie on the real line. It turns out that the polynomial $P_n$, for any $n \geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized eigenvalue problem. It is shown that with th...

We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple and lie on the real line. It turns out that the polynomial $P_n$, for any $n \geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized eigenvalue problem. It is shown that with th...

We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometr...

We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometr...

We prove that the function (Formula presented.) and its (Formula presented.)-analogue are of the form (Formula presented.) and (Formula presented.) is completely monotonic in (Formula presented.). In particular both (Formula presented.) and (Formula presented.) are Laplace transforms of infinitely divisible distributions. We also extend Lerch’s ine...

We provide a complete spectral analysis of all self-adjoint operators acting on $\ell^{2}(\mathbb{Z})$ which are associated with two doubly infinite Jacobi matrices with entries given by $$ q^{-n+1}\delta_{m,n-1}+q^{-n}\delta_{m,n+1} $$ and $$ \delta_{m,n-1}+\alpha q^{-n}\delta_{m,n}+\delta_{m,n+1}, $$ respectively, where $q\in(0,1)$ and $\alpha\in...

We provide a complete spectral analysis of all self-adjoint operators acting on $\ell^{2}(\mathbb{Z})$ which are associated with two doubly infinite Jacobi matrices with entries given by $$ q^{-n+1}\delta_{m,n-1}+q^{-n}\delta_{m,n+1} $$ and $$ \delta_{m,n-1}+\alpha q^{-n}\delta_{m,n}+\delta_{m,n+1}, $$ respectively, where $q\in(0,1)$ and $\alpha\in...

We derive two new versions of Cooper's formula for the iterated Askey–Wilson operator. Using the second version of Cooper's formula and the Leibniz rule for the iterated Askey–Wilson operator, we derive several formulas involving this operator. We also give new proofs of Rogers' summation formula for series, Watson's transformation, and we establis...

By applying an integral representation for $q^{k^{2}}$ we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of $q$-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problem...

By applying an integral representation for $q^{k^{2}}$ we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of $q$-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problem...

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.

We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by Ramanujan, as well as certain identities with an interesting variable-parameter symmetry based on limiting cases...

We evaluate $q$-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the $m$-version of the Rogers-Ramanujan identities. We also prove several generating functions for Stieltjes-Wigert polynomials with argument depending on the degree. In addition we give several Rogers...

We evaluate $q$-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the $m$-version of the Rogers-Ramanujan identities. We also prove several generating functions for Stieltjes-Wigert polynomials with argument depending on the degree. In addition we give several Rogers...

We prove a generalization of the Kibble-Slepian formula (for Hermite polynomials) and its unitary analogue involving the $2D$ Hermite polynomials recently proved in \cite{Ism4}. We derive integral representations for the $2D$ Hermite polynomials which are of independent interest. Several new generating functions for $2D$ $q$-Hermite polynomials wil...

In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation...

The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah-Wilson algebra associated to the generic Racah/Wils...

The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah-Wilson algebra associated to the generic Racah/Wils...

We study the complex Hermite polynomials {H-m,H- n(z, (z) over bar)} in some detail, establish operational formulas for them and prove a Kibble-Slepian type formula, which extends the Poisson kernel for these polynomials. Positivity of the associated kernels is discussed. We also give an infinite family of integral operators whose eigenfunctions ar...

We consider three different systems of dual q-integral equations where the kernel is the third Jackson q-Bessel functions. We solve the first system by applying the multiplying factor method (ansatz solution) and the second by employing the fractional q-calculus, and we use the q-Mellin transform to reduce the third system to a Fredholm q-integral...

We give a general expansion formula of functions in the Askey–Wilson polynomials and using Askey–Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey–Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q-Jacobi type polyn...

This work contains a detailed study of a one parameter generalization of the 2D-Hermite polynomials and a two parameter extension of Zernike's disc polynomials. We derive linear and bilinear generating functions, and explicit formulas for our generalizations and study integrals of products of some of these 2D orthogonal polynomials. We also establi...

The first author has recently proved a Kibble–Slepian type formula for the 2D-Hermite polynomials which extends the Poisson kernel for these polynomials. We provide a combinatorial proof of a closely related formula. The combinatorial structures involved are the so-called m-involutionary ℓ-graphs. They are enumerated in two different manners: first...

We derive inequalities and a complete asymptotic expansion for the Landau constants G(n), as n -> infinity using the asymptotic sequence n!/(n + k)!. We also introduce a q-analogue of the Landau constants and calculate their large degree asymptotics. In the process, we also establish q-analogues of identities due to Ramanujan and Bailey.

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-$D$ Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give $q$-ana...

We propose a novel random transfer matrix for quantum transport in disordered systems. The model is exactly solvable in the sense that arbitrary n-point correlation functions of the eigenvalues can be obtained from known orthogonal polynomials, and the conductance is a simple linear statistics of these eigenvalues.

We introduce two q-analogues of the 2D-Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues formulas for both families. We also introduce a q-2D analogue of the disk polynomials (Zernike polynomials) and de...

We study a class of bivariate deformed Hermite polynomials and some of their properties using classical analytic techniques and the Wigner map. We also prove the positivity of certain determinants formed by the deformed polynomials. Along the way we also work out some additional properties of the (undeformed) complex Hermite polynomials and their r...

We establish integral representations of Heine type for certain integrals of determinants. We use these representations to derive monotonicity properties of such determinants. New integral representations of Heine type for biorthogonal functions obtained from the general Gram–Schmidt orthonormalization process are given. We also establish the monot...

We consider two types of Hermite polynomials of a complex variable. For each type we obtain combinatorial interpretations for the linearization coefficients of products of these polynomials. We use the combinatorial interpretations to give new proofs of several orthogonality relations satisfied by these polynomials with respect to positive exponent...

We give new derivations of properties of the functions of the second kind of the Jacobi, little and big q-Jacobi polynomials, and the symmetric Al-Salam–Chihara polynomials for q>1q>1. We also study the Askey–Wilson functions and the Wilson functions of second kind. An integration by parts formula is derived for the Wilson operator in appropriate H...

We study the asymptotic behavior of Laguerre polynomials Ln(αn)(z) as n→∞n→∞, where αn/nαn/n has a finite positive limit or the limit is +∞+∞. Applying the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we derive the uniform asymptotics of such polynomials, which improves on the results of Bosbach and Gawronski (1998). I...

We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure dμ(x)dμ(x), supported on the interval (a,b)(a,b) and the other with respect to the measure |x−c|τ|x−d|γdμ(x)|x−c|τ|x−d|γdμ(x), where cc and dd are outside (a,b)(a,b). We prove that the zeros of the...

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials. This shows that co-recursive polynomials may lead to interesting sets of orthogonal polynomials.

We derive the asymptotics of certain combinatorial numbers defined on multi-sets when the number of sets tends to infinity but the sizes of the sets remain fixed. This includes the asymptotics of generalized derangements, numbers related to k-partite graphs, and exponentially weighted derangements. The asymptotics use integral and sum representatio...

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra gene...

The class of hypergeometric polynomials F-2(1) (-m, b; b + (b) over bar; 1 - z) with respect to the parameter b = lambda + i eta, where lambda > 0, are known to have all their zeros simple and exactly on the unit circle vertical bar z vertical bar = 1. In this note we look at some of the associated extremal and orthogonal properties on the unit cir...

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger-Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a...

We study the Plancherel--Rotach asymptotics of four families of orthogonal polynomials, the Chen--Ismail polynomials, the Berg-Letessier-Valent polynomials, the Conrad--Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems and three of them are birth and death process polynomials with cubic or quartic rates. We...

We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q < 1$. Each of these results implies the infinite divisibility of a related probability measure. In a few cases, we are able to get simple monotonicity without having complete monotonicity. All of...

The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos \theta$, which are given as a terminating $_4\phi_3$ basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in $z=e^{i\theta}$, which are given as a sum of two terminating $_4\phi_3$'s. They satisfy a biorthogonality relation. In this paper...

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which...

We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam–Chihara, Askey–Wilson, and Meixner–Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obe...

A general scheme for tridiagonalizing differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson...

The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with expl...

We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q^{-1}-Hermite polynomials of Askey, Ismail and Masson, q-La...

The $J$-matrix method is extended to difference and $q$-difference operators and is applied to several explicit differential, difference, $q$-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most case...

We obtain a symmetric tridiagonal matrix representation of the Dirac-Coulomb operator in a suitable complete square integrable basis. Orthogonal polynomials techniques along with Darboux method are used to obtain the bound states energy spectrum, the relativistic scattering amplitudes and phase shifts from the asymptotic behavior of the polynomial...

We analyze the degree of shape preserving weighted polynomial approximation for exponential weights on the whole real line. In particular, we establish a Jackson type estimate.

In this chapter we introduce the study held by Annaby and Mansour in (J. Phys. A Math. Gen. 38(17), 3775-3797, 2005) of a self adjoint basic Sturm-Liouville eigenvalue problem in a Hilbert space. The last two sections of this chapter are about the q(2)-Fourier transform introduced by Rubin in (J. Math. Anal. Appl. 212(2), 571-582, 1997; Proc. Am. M...

In this chapter we investigate q-analogues of the classical fractional calculi. We study the q-Riemann-Liouville fractional integral operator introduced by Al-Salam (Proc. Am. Math. Soc. 17, 616-621, 1966; Proc. Edinb. Math. Soc. 2(15), 135-140, 1966/1967) and by Agarwal (Proc. Camb. Phil. Soc. 66, 365370, 1969). We give rigorous proofs of existenc...