a relativistic hypergeometric function we survey our work on a function generalizing 2f1 this function is a joint eigenfunction of four askey wilson type hyperbolic difference operators reducing to the askey wilson polynomials for certain discrete values of the variables it is defined by a contour integral generalizing the barnes representation of 2f1 it has various symmetries including a hidden d4 symmetry in the parameters by means of the associated hilbert space transform the difference operators can be promoted to self adjoint operators provided the parameters vary over a certain polytope in the parameter space for a dense subset of parameter shifts give rise to an explicit evaluation in terms of rational functions of exponentials hyperbolic functions and plane waves

special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

new bounds for the extreme values of a finite sample of real numbers we derive new bounds for the smallest valuexminand the largest valuexmaxof a finite samplex1 xnof real numbers our bounds are obtained by solving two optimization problems one of them being convex and the other nonconvex we show that the pair xmin xmax lies in a region bounded by an ellipse and an hyperbola the corresponding cartesian equations are given in terms of the average and the standard deviation of the sample

on some new contiguous relations for the gauss hypergeometric function with applications contiguous relations for hypergeometric series contain an enormous amount of hidden information applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series in this paper a new set of contiguous function relations are established applications of such relations to hypergeometric summation formulas and the theory of jacobi polynomials are presented

bispectral extensions of the askey wilson polynomials following the pioneering work of duistermaat and grunbaum we call a family pn x n 0 of polynomials bispectral if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators one consisting of difference operators acting on the degree index n and another one of operators acting on the variable x the goal of the present paper is to construct and parametrize bispectral extensions of the askey wilson polynomials where the second algebra consists of q difference operators in particular we describe explicitly measures on the real line for which the corresponding orthogonal polynomials satisfy higher order q difference equations extending all known families of orthogonal polynomials satisfying q difference difference or differential equations in x

extensions of discrete classical orthogonal polynomials beyond the orthogonality abstract it is well known that the family of hahn polynomials h n x n n 0 is orthogonal with respect to a certain weight function up to degree n in this paper we prove by using the three term recurrence relation which this family satisfies that the hahn polynomials can be characterized by a sobolev orthogonality for every n and present a factorization for hahn polynomials for a degree higher than n we also present analogous results for dual hahn krawtchouk and racah polynomials and give the limit relations among them for all n n 0 furthermore in order to get these results for the krawtchouk polynomials we will obtain a more general property of orthogonality for meixner polynomials

polynomials and operator orderings it is shown that there is an exact one to one correspondence between all possible sets of polynomials both orthogonal and nonorthogonal and rules for operator orderings operator orderings that are hermitian give polynomials with definite parity most of the standard classical orthogonal polynomials are associated with operator orderings that are not particularly simple however there is a special one parameter class of hermitian operator orderings that corresponds to a class of elegant but little known orthogonal polynomials called continuous hahn polynomials

spectral decomposition and matrix valued orthogonal polynomials 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 poly nomials with explicit orthogonality relations and three term recurrence relation are pre sented which both can be considered as 2 2 matrix valued analogues of subfamilies of askey wilson polynomials

intertwiners of uq sl 2 representations and the vector valued big q jacobi transform linear operators r are introduced on tensor products of evaluation modules of uq sl 2 obtained from the complementary and strange series representations the operators r satisfy the intertwining condition on finite linear combinations of the canonical basis elements of the tensor products infinite sums associated with the action of r on six pairs of tensor products are evaluated for two pairs the sums are related to the vector valued big q jacobi transform of the matrix elements defining the operator r in one case the sums specify the action of r on the irreducible representations present in the decomposition of the underlying indivisible sum of uqsl 2 tensor products in both cases bilinear summation formulae for the matrix elements of r provide a generalization of the unitarity property corresponding results are given for the remaining pairs

a logarithmic barrier approach for linear programming this paper presents a logarithmic barrier method for solving a linear programming problem we are interested in computation of the direction by the newtons method and in computation of the displacement step using majorant functions instead line search methods in order to reduce the computation cost this purpose is confirmed by numerical experiments showing the efficiency of our approach which are presented in the last section of this paper

properties of generalized univariate hypergeometric functions based on spiridonov s analysis of elliptic generalizations of the gauss hypergeometric function we develop a common framework for 7 parameter families of generalized elliptic hyperbolic and trigonometric univariate hypergeometric functions in each case we derive the symmetries of the generalized hypergeometric function under the weyl group of type e 7 elliptic hyperbolic and of type e 6 trigonometric using the appropriate versions of the nassrallah rahman beta integral and we derive contiguous relations using fundamental addition formulas for theta and sine functions the top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with ruijsenaars relativistic hypergeometric function and the askey wilson function respectively we show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions we also describe an intimate connection between the hyperbolic and trigonometric theory which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions

multivariable continuous hahn polynomials a multivariable generalization of the continuous hahn polynomials is presented it is a 4p 4 parameter family where p is the number of variables it is shown that they are orthogonal with respect to subspaces of equal degree and biorthogonal within a given subspace in the simplest case the multivariable weight function takes the form sech x1 x2 xp sech x1 sech x2 sech xp

meixner pollaczek polynomials and the heisenberg algebra an alternative proof is given for the connection between a system of continuous hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender mead and pinsky phys rev lett 56 2445 1986 j math phys 28 509 1987 the continuous hahn polynomials turn out to be meixner pollaczek polynomials use is made of the connection between laguerre polynomials and meixner pollaczek polynomials the rodrigues formula for laguerre polynomials an operational formula involving meixner pollaczek polynomials and the schrodinger model for the irreducible unitary representations of the three dimensional heisenberg group

ramanujan type continuous measures for biorthogonal q rational functions it is shown that ramanujan type continuous measures for a hierarchy of biorthogonal rational functions can be systematically built from simple cases of the rogers szego for 0 1 polynomials by using the berg ismail procedure of attaching generating functions to measures

an optimized wave equation for seismic modeling and reverse time migration the acoustic wave equation has been widely used for the modeling and reverse time migration of seismic data numerical implementation of this equation via finite difference techniques has established itself as a valuable approach and has long been a favored choice in the industry to ensure quality results accurate approximations are required for spatial and time derivatives traditionally they are achieved numerically by using either relatively very fine computation grids or very long finite difference operators otherwise the numerical error known as numerical dispersion is present in the data and contaminates the signals however either approach will result in a considerable increase in the computational cost a simple and computationally low cost modification to the standard acoustic wave equation is presented to suppress numerical dispersion this dispersion attenuator is one analogy of the antialiasing operator widely applied in kirchhoff migration when the new wave equation is solved numerically using finite difference schemes numerical dispersion in the original wave equation is attenuated significantly leading to a much more accurate finite difference scheme with little additional computational cost numerical tests on both synthetic and field data sets in both two and three dimensions demonstrate that the optimized wave equation dramatically improves the image quality by successfully attenuating dispersive noise the adaptive application of this new wave equation only increases the computational cost slightly

ordered products w algebra and two variable definite parity orthogonal polynomials it has been shown that the cartan subalgebra of w algebra is the space of the two variable definite parity polynomials explicit expressions of these polynomials and their basic properties are presented it also has been shown that they carry the infinite dimensional irreducible representation of the su 1 1 algebra having the spectrum bounded from below a realization of this algebra in terms of difference operators is also obtained for particular values of the ordering parameter s they are identified with the classical orthogonal polynomials of a discrete variable such as the meixner meixner pollaczek and askey wilson polynomials with respect to variable s they satisfy a second order eigenvalue equation of hypergeometric type exact scattering states with zero energy for a family of potentials are expressed in terms of these polynomials it has been put forward that it is the inonu wigner contraction and its inverse that form a bridge between the difference and differential calculus

contiguous relations basic hypergeometric functions and orthogonal polynomials iii associated continuous dual q hahn polynomials 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

extension of a projective interior point method for linearly constrained convex programming in this paper we propose an interior point method of type projective to minimize a nonlinear function under linear constraints we combine the approach of linearization with ingredients brought by karmarkar the theoretical results deduct of those established later

theq laguerre polynomials and related moment problems abstract we study two indeterminate hamburger moment problems associated with q laguerre polynomials the coefficients in their recurrence relations are of exponential growth this completes earlier work started by d moak some of the measures that the polynomials are orthogonal with respect to are found and lead to the evaluation of certain integrals a symmetric set of orthogonal polynomials is also considered

confluent hypergeometric orthogonal polynomials related to the rational quantum calogero system with harmonic confinement two families type a and type b of confluent hypergeometric polynomials in several variables are studied we describe the orthogonality properties differential equations and pieri type recurrence formulas for these families in the one variable case the polynomials in question reduce to the hermite polynomials type a and the laguerre polynomials type b respectively the multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum calogero models with harmonic confinement for the classical root systems and are closely connected to the symmetric generalized spherical harmonics investigated by dunkl

implicit finite difference simulations of seismic wave propagation we propose a new finite difference modeling method implicit both in space and in time for the scalar wave equation we use a three level implicit splitting time integration method for the temporal derivative and implicit finite difference operators of arbitrary order for the spatial derivatives both the implicit splitting time integration method and the implicit spatial finite difference operators require solving systems of linear equations we show that it is possible to merge these two sets of linear systems one from implicit temporal discretizations and the other from implicit spatial discretizations to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm we give the complete derivations of the implicit splitting time integration method and the implicit spatial finite difference operators and present the resulting discretized formulas for the scalar wave equation we conduct a thorough numerical analysis on grid dispersions of this new implicit

a queueing model and a set of orthogonal polynomials a single serving queueing model is studied where potential customers are discouraged at the rate n qn 0 q 1 n is the queue length the serving rate is n 1 qn n 0 1 the spectral function is computed and the corresponding set of orthogonal polynomials is studied in detail the slightly more general model with n qn 1 bqn n 1 qn 1 bqn and the analogous orthogonal polynomials are also investigated in both cases a method developed by pollaczek is used which has been used very successfully to study new sets of orthogonal polynomials by askey and ismail

some q beta and mellin barnes integrals on compact lie groups and lie algebras multidimensional generalizations of beta type integrals of bames ramanujan askey wilson and others are evaluated these integrals are analogues of the summation theorems for multilateral hypergeometric series associated to the simple lie algebras of classical type and type g2 many of these integrals can also be written as group integrals over a compact lie group or conjugation invariant integrals over the corresponding lie algebra

contiguous relations basic hypergeometric functions and orthogonal polynomials i abstract the second order q difference equation of a 2 1 and the contiguous relations of 2 1 s are used to derive continued fraction expansions for quotients of 2 1 s we identify the cases when these continued fractions are positive definite j fractions i e the cases when the denominators of the approximants of the j fractions are orthogonal polynomials the measures with respect to which the denominator polynomials are orthogonal are found and the orthogonal polynomials are studied in some detail

contiguous relations basic hypergeometric functions and orthogonal polynomials ii associated big q jacobi polynomials abstract explicit solutions to the recurrence relation for the associated big q jacobi polynomials are obtained using 3 2 contiguous relations a new continued fraction is derived with the help of the minimal solution the orthogonality for polynomial solutions in some special cases is discussed two exceptional cases yield the basic analogues of entry 25 and entry 33 in chapter 12 of ramanujan s second notebook connections of the general case with other associated cases are indicated

an extension of karmarkar projective algorithm for convex quadratic programming we present an extension of karmarkar s linear programming algorithm for solving a more general group of optimization problems convex quadratic programs this extension is based on the iterated application of the objective augmentation and the projective transformation followed by optimization over an inscribing ellipsoid centered at the current solution it creates a sequence of interior feasible points that converge to the optimal feasible solution in o ln iterations each iteration can be computed in o ln3 arithmetic operations wheren is the number of variables andl is the number of bits in the input in this paper we emphasize its convergence property practical efficiency and relation to the ellipsoid method

contiguous relations and their computations for f12 hypergeometric series the hypergeometric function f12 a 1 a 2 a 3 z plays an important role in mathematical analysis and its application gauss defined two hypergeometric functions to be contiguous if they have the same power series variable if two of the parameters are pairwise equal and if the third pair differs by 1 he showed that a hypergeometric function and any two other contiguous to it are linearly related in this paper we present an interesting formula as a linear relation of three shifted gauss polynomials in the three parameters a 1 a 2 and a 3 more precisely we obtained a recurrence relation including f12 a email protected 1 a 2 a 3 z f12 a 1 a email protected 2 a 3 z andf12 a 1 a 2 a email protected 3 z for any arbitrary integers a 1 a 2 and a 3

an extended variant of karmarkar s interior point algorithm abstract in this paper extended algorithm of karmarkar s interior point algorithm is considered theoretical aspects of the new algorithm are discussed we show that the new iterative algorithm converges faster than former karmarkar s algorithms successful convergence for problems of different sizes is obtained numerical results show that when we use a new parameter in the classical karmarkar s algorithm number of iterations is less than the number of iterations for two mentioned methods

diagonalizaton of certain integral operators ii abstract we establish an integral representation of a right inverse 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

a stability formula for lax wendroff methods with fourth order in time and general order in space for the scalar wave equation based on the formula for stability of finite difference methods with second order in time and general order in space for the scalar wave equation i obtain a stability formula for lax wendroff methods with fourth order in time and general order in space unlike the formula for methods with second order in time this formula depends on two parameters one parameter is related to the weights for approximations of second spatial derivatives the other parameter is related to the weights for approximations of fourth spatial derivatives when discretizing the mixed derivatives properly the formula can be generalized to the case where the spacings in different directions are different this formula can be useful in high accuracy seismic modeling using the scalar wave equation on rectangular grids which involves both high order spatial discretizations and high order temporal approximations i also prove the instability of methods obtained by applying high order finite difference approximations directly to the second temporal derivative and this result solves the bording s conjecture

a recipe for stability of finite difference wave equation computations finite difference solutions to the wave equation are pervasive in the modeling of seismic wave propagation kelly and marfurt 1990 and in seismic imaging bording and lines 1997 that is they are useful for the forward problem modeling and the inverse problem migration in computational solutions to the wave equation it is necessary to be aware of conditions for numerical stability in this short note we examine a convenient recipe for insuring stability in our finite difference solutions to the wave equation the stability analysis for finite difference solutions of partial differential equations is handled using a method originally developed by von neumann and described by press et al 1986 p 827 830

the rotating harmonic oscillator eigenvalue problem i continued fractions and analytic continuation the continued fraction approach to the solution of the rotating harmonic oscillator eigenvalue problem is examined in detail it is shown how one may obtain eigenvalue information only from an analytic continuation of the continued fraction accomplished with the aid of modified approximants

multivariable biorthogonal hahn polynomials a multivariable biorthogonal generalization of the discrete hahn polynomials a p 1 complex parameter family where p is the number of variables is presented it is shown that the polynomials are orthogonal with respect to subspaces of lower degree and biorthogonal within a given subspace these properties are over the discrete simplex 0 x1 x2 xp where x1 x2 xp and are non negative integers some further properties of the closely related multivariable continuous hahn polynomials are also discussed

tensor products of uq sl 2 modules and the big q2 jacobi function transform four tensor products of evaluation modules of the quantum affine algebra uq sl 2 obtained from the negative and positive series the complementary and the strange series representations are investigated linear operators r z satisfying the intertwining property on finite linear combinations of the canonical basis elements of the tensor products are described in terms of two sets of infinite sums r t r t z 0 and r t r t z 0 involving big q2 jacobi functions or related nonterminating basic hypergeometric series inhomogeneous recurrence relations can be derived for both sets evaluations of the simplest sums provide the corresponding initial conditions for the first set of sums the relations entail a big q2 jacobi function transform pair an integral decomposition is obtained for the sum r t a partial description of the relation between the decompositions of the tensor products with respect to uqsl 2 or with respect to its complement in uq sl 2 can be formulated in terms of askey wilson

continuous hahn polynomials and the heisenberg algebra continuous hahn polynomials have surfaced in a number of somewhat obscure physical applications for example they have emerged in the description of two photon processes in hydrogen hard hexagon statistical mechanical models and clebsch gordan expansions for unitary representations of the lorentz group so 3 1 in this paper it is shown that there is a simple and elegant way to construct these polynomials using the heisenberg algebra

lowrank seismic wave extrapolation on a staggered grid we evaluated a new spectral method and a new finite difference fd method for seismic wave extrapolation in time using staggered temporal and spatial grids we derived a wave extrapolation operator using a lowrank decomposition for a first order system of wave equations and designed the corresponding fd scheme the proposed methods extend previously proposed lowrank and lowrank fd wave extrapolation methods from the cases of constant density to those of variable density dispersion analysis demonstrated that the proposed methods have high accuracy for a wide wavenumber range and significantly reduce the numerical dispersion the method of manufactured solutions coupled with mesh refinement was used to verify each method and to compare numerical errors tests on 2d synthetic examples demonstrated that the proposed method is highly accurate and stable the proposed methods can be used for seismic modeling or reverse time migration

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

the hahn and meixner polynomials of an imaginary argument and some of their applications the hahn and meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter this leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure the meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the pollaczek polynomials the orthogonality relation for the hahn polynomials with respect to a continuous measure is apparently new a close connection between the hahn polynomials of an imaginary argument and representations of the lorentz group so 3 1 is considered

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

multivariable continuous hahn and wilson polynomials related to integrable difference systems multivariable generalizations of the continuous hahn and wilson polynomials are introduced as eigenfunctions of rational ruijsenaars type difference systems with an external field

zeros and orthogonality of the askey wilson polynomials for q a root of unity we study some properties of the askey wilson polynomials awp when q is a primitive nth root of unity for general four parameter awp zeros of the nth polynomial and the orthogonality measure are found explicitly special subclasses of the awp e g the continuous q jacobi and big q jacobi polynomials are considered in detail a set of discrete weight functions positive on a real interval is described some new trigonometric identities related to the awp are obtained normalization conditions of some polynomials are expressed in terms of the gauss sums

fourier finite difference wave propagation we introduce a novel technique for seismic wave extrapolation in time the technique involves cascading a fourier transform operator and a finite difference operator to form a chain operator fourier finite differences ffd we derive the ffd operator from a pseudoanalytical solution of the acoustic wave equation two dimensional synthetic examples demonstrate that the ffd operator can have high accuracy and stability in complex velocity media applying the ffd method to the anisotropic case overcomes some disadvantages of other methods such as the coupling of qp waves and qsv waves the ffd method can be applied to enhance accuracy and stability of seismic imaging by reverse time migration

eigenvalue integrodifferential equations for orthogonal polynomials on the real line the one dimensional harmonic oscillator wave functions are solutions to a sturm liouville problem posed on the whole real line this problem generates the hermite polynomials however no other set of orthogonal polynomials can be obtained from a sturm liouville problem on the whole real line in this paper it is shown how to characterize an arbitrary set of polynomials orthogonal on in terms of a system of integrodifferential equations of the hartree fock type this system replaces and generalizes the linear differential equation associated with a sturm liouville problem our results are demonstrated for the special case of continuous hahn polynomials

analysis of higher order finite difference schemes in 3 d reverse time migration the design and usefulness of practical algorithms for 3 d reverse time depth migration is examined while demonstrating data applications of the algorithm we evaluate quantitatively the accuracy of the finite difference operator from second order to eighth order for the scalar wave equation by comparing numerical and analytical solutions the results clearly show the advantage of using higher order finite difference schemes especially from second order to fourth order for space derivatives hence a finite difference method with the accuracy of fourth order in space and second order in time is applied to 3 d full scalar wave equations in reverse time migration considerable savings in cpu and memory are obtained by using larger horizontal grid spacings than the vertical spacings therefore we derive dispersion and stability conditions for such unequal grid spacing the 3 d reverse time migration of data from the hibernia field shows an image improvement over 2 d migrations despite the fact that much of the structure was considered to be approximately 2 d stable nondispersive finite difference methods of higher order can allow for tractable and efficient 3 d reverse time migration solutions

multivariable meixner krawtchouk and meixner pollaczek polynomials a multivariable biorthogonal generalization of the meixner krawtchouk and meixner pollaczek polynomials is presented it is shown that these are orthogonal with respect to subspaces of lower degree and biorthogonal within a given subspace the weight function associated with the krawtchouk polynomials is the multivariate binomial distribution

properties of some families of hypergeometric orthogonal polynomials in several variables limiting cases are studied of the koornwinder macdonald multivariable generalization of the askey wilson polynomials we recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable wilson continuous hahn and jacobi type polynomials respectively for each class of polynomials we provide systems of difference or differential equations recurrence relations and expressions for the norms of the polynomials in terms of the norm of the constant polynomial

the rotating harmonic oscillator eigenvalue problem ii analytic perturbation theory the theory of self adjoint analytic families is applied to the rotating harmonic oscillator hamiltonian in l2 0 to obtain weak and strong coupling expansions of the eigenvalues various estimates on the radius of convergence of the weak coupling expansion are obtained the strong coupling expansion is shown to be an asymptotic series which with the neglect of exponentially small terms is expressible in terms of a simple formal perturbation of the ordinary harmonic oscillator hamiltonian in l2

tribasic integrals and identities of rogers ramanujan type some integrals involving three bases are evaluated as infinite products using complex analysis many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products the resulting identities are identities of rogers ramanujan type some integer partition interpretations of these identities are given generalizations of the rogers ramanujan type identities involving polynomials are given again as corollaries of integral evaluations

discrete time quantum mechanics abstract this paper summarizes a research program that has been underway for a decade the objective is to find a fast and accurate scheme for solving quantum problems which does not involve a monte carlo algorithm we use an alternative strategy based on the method of finite elements we are able to formulate fully consistent quantum mechanical systems directly on a lattice in terms of operator difference equations one advantage of this discretized formulation of quantum mechanics is that the ambiguities associated with operator ordering are eliminated furthermore the scheme provides an easy way in which to obtain the energy levels of the theory numerically a generalized version of this discretization scheme can be applied to quantum field theory problems the difficulties normally associated with fermion doubling are eliminated also one can incorporate local gauge invariance in the finite element formulation results for some field theory models are summarized in particular we review the calculation of the anomaly in two dimensional quantum electrodynamics the schwinger model finally we discuss nonabelian gauge theories

continuous hahn polynomials a slightly more general orthogonality relation for the hahn polynomials of a continuous variable than the recent one given by atakishiyev and suslov 1985 is given here

a multidimensional generalization of the barnes integral and the continuous hahn polynomial employing a multidimensional generalization of the beta integral of barnes we derive an expression of the continuous hahn polynomials motivation of the present work is to study such integrals from the viewpoint of a finite difference version of the de rham theory

canonical commutation relation preserving maps we study maps preserving the heisenberg commutation relation ab ba 1 we find a one parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative it involves a non local coordinate operator while the dual derivative is just the jackson finite difference operator substitution of this realization into any differential operator involving x and d dx results in an isospectral deformation of a continuous differential operator into a finite difference one we extend our results to the deformed heisenberg algebra ab qba 1 as an example of potential applications various deformations of the hahn polynomials are briefly discussed

determination of finite difference weights using scaled binomial windows the finite difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes conventional finite difference weights can be calculated either from taylor series expansions or by lagrange interpolation polynomials the finite difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain for this reason we also can derive finite difference operators by truncating the convolution series of the pseudospectral method various truncation windows can be employed for this purpose and they result in finite difference operators with different dispersion properties we found that there exists two families of scaled binomial windows that can be used to derive conventional finite difference operators analytically with a minor change these scaled binomial windows can also be used to derive optimized finite difference operators with enhanced dispersion properties

big q laguerre and q meixner polynomials and representations of the quantum algebra uq su1 1 diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra uq su1 1 is studied spectrum and eigenfunctions of this operator are explicitly found these eigenfunctions when normalized constitute an orthonormal basis in the representation space the initial uq su1 1 basis and the basis of these eigenfunctions are interconnected by a matrix with entries expressed in terms of big q laguerre polynomials the unitarity of this connection matrix leads to an orthogonal system of functions which are dual with respect to big q laguerre polynomials this system of functions consists of two separate sets of functions which can be expressed in terms of q meixner polynomials mn x b c q either with positive or negative values of the parameter b the orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix as a consequence one obtains an orthogonality relation for the q meixner polynomials mn x b c q with b 0 a biorthogonal system of functions with respect to the scalar product in the representation space is also derived