arXiv subjects
Publications and source records attributed to Mourad E. H. Ismail.
We explore in this paper some orthogonal polynomials which are naturally associated to certain families of coherent states, often referred to as nonlinear coherent states in the quantum optics literature. Some examples turn out to be known orthogonal polynomials but in many cases we encounter a general class of new orthogonal polynomials for which we establish several qualitative results.
A general scheme for tridiagonalising 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 polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey-Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).
Given an operator L acting on a function space, the J-matrix method consists of finding a sequence y_n of functions such that the operator L acts tridiagonally on y_n with respect to n. Once such a tridiagonalization is obtained, a number of characteristics of such an operator L can be obtained. In particular, information on eigenvalues and eigenfunctions, bound states, spectral decompositions, etc. can be obtained in this way. We review the general set-up, and we discuss two examples in detail; the Schrodinger operator with Morse potential and the Lame equation.
The "Diophantine" property of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, is explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising from the basic hypergeometric series. We found, with suitably chosen parameters and certain $q-$analogue of the summation theorems, zeros of these polynomials explicitly, which are no longer integer valued. This goes beyond the results obtained by the Authors mentioned above.
The q-difference analog of the classical ladder operators is derived for those orthogonal polynomials arising from a class of indeterminate moments problem.
We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given .
We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a new scaling appropriate for such power law densities (different from the scaling required in Gaussian random matrix ensembles), we calculate exactly the two-level kernel that determines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles.
We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path integral representation. This is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that works over the full range of positions, $n,$ including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of this system have run into difficulties in the transitional range, because the approximations on which they were based break down here. The fact that there are two different kinds of approach to this problem (Path Integral vs. Schrödinger wave mechanics) is ultimately a manifestation of the equivalence between the path-integral formulation of quantum mechanics and the original formulation developed in the 1920s. We discuss how and why our approach is related to the two methods that have already been used to analyse these systems.
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.
In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find expresions for this lower bound in a number of indeterminate moment problems.
Explicit solutions for the three-term recurrence satisfied by associated continuous dual $q$-Hahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure is obtained. Eleven limit cases are discussed in some detail. These include associated big $q$-Laguerre , associated Wall, associated Al-Salam-Chihara, associated Al-Salam-Carlitz I, and associated continuous $q$-Hermite polynomials.
We find the adjoint of the Askey-Wilson divided difference operator with respect to the inner procuct on L^2(-1,1,(1-x^2)^-1/2 dx) defined as a Cauchy principle value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.
The connection between continued fractions and orthogonality which is familiar for $J$-fractions and $T$-fractions is extended to what we call $R$-fractions of type I and II. These continued fractions are associated with recurrence relations that correspond to multipoint rational interpolants. A Favard type theorem is proved for each type. We then study explicit models which lead to biorthogonal rational functions.
We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to ${}_4ϕ_3$ biorthogonal rational functions on the unit circle.
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
By splitting the real line into intervals of unit length a doubly infinite integral of the form $\Int F(q^x)\,dx,\; 0<q<1$, can clearly be expressed as $\Integ \Sum F(q^{x+n})\,dx$, provided $F$ satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his \ph{1}{1} sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised \ph{2}{2} sum as well as the very-well-poised \ph{6}{6} sum are also found in a straightforward manner. An extension to a very-well-poised and balanced \ph{8}{8} series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.
We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on $L^2$ with weight $(1-x^2)^{-1/2}$. The kernel of this integral operator is $\vartheta'_4/\vartheta_4$ and is the Riemann mapping function that maps the open unit disc conformally onto the interior of an ellipse.
We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an $L^2$ space weighted by the weight function of the continuous $q$-Jacobi polynomials. We characterize the eigenvalues of this integral operator and prove a $q$-analog of the expansion of $e^{ixy}$ in Jacobi polynomials of argument $x$. We also outline a general procedure of finding integral representations for inverses of linear operators.