the laguerre and legendre polynomials from an operational point of view the use of the monomiality principle and of the associated operational calculus allows to reformulate the theory of laguerre and legendre polynomials from a unified point of view they are indeed shown to be particular cases of more general polynomials whose usefulness in purely mathematical and applied contexts is discussed the concepts introduced in the paper are also exploited to develop an elementary point of view on the theory of legendre and laguerre bessel functions

generalized hermite polynomials associated with functions of parabolic cylinder associated hermite polynomials he n n z which generalize the usual scaled hermite polynomials he n z corresponding to n 0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder d n z in finite integer steps properties of these polynomials such as recursion relations explicit representations and the differential equation are derived the generation of the associated hermite polynomials from the usual hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given an application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder

integral transforms and chebyshev like polynomials we use the integral representation method to derive the properties of chebyshev polynomials and of their generalized forms

computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method we propose in this paper a 1d model problem to study the convergence of surrogate approximations using an atomistic to continuum coupling method towards the solution of a full particle model the 1d problem consists of a collection of springs that exhibits a local defect materialized by a sudden change in the spring properties the surrogate model is obtained by the arlequin approach which introduces an overlap region in which the continuum and particle models are coupled together using lagrange multipliers the objective of the present work is to show via numerical experiments that the modeling error does indeed converge to zero as the distance of the overlap region from the defect and or its size are increased

associated laguerre konhauser polynomials quasi monomiality and operational identities abstract in this paper we exploit the monomiality principle to discuss and introduce a new class of laguerre konhauser polynomials we study their properties differential equations generating functions recurrence relations expansions and so on and discuss the link with ordinary case

about macroscopic models of instability pattern formation abstract various macroscopic models to describe instability pattern formation are discussed in this paper they are similar to the ginzburg landau envelope equation but they can remain valid away from the bifurcation and are based on the technique of fourier series with slowly varying coefficients we focus on two questions the need to take phase changes into account and the boundary conditions to be associated with macroscopic models the analysis is carried out on the basis of numerical simulations for the problem of a compressed beam on a nonlinear foundation that is quite similar to the well known swift hohenberg equation the first macroscopic model involves a real envelope so that the phase is assumed to be constant the second model is also macroscopic and it is a sort of ginzburg landau equation with a complex envelope the third one follows from a multi scaled approach with a numerical bridging between the full model near the boundary and a macroscopic model in the bulk

operational rules and arbitrary order hermite generating functions operational identities often exploited in the theory of exponential operators are used to derive generating functions of hermite and kampe de feriet polynomials the method we propose allows us to obtain unilateral bilinear and bilateral generating functions by using the same procedure possible extensions of the technique are also discussed

operational methods and differential equations with applications to initial value problems the authors present and develop a general method of operational nature in order to explore and investigate such families of orthogonal polynomials and functions as the classical and generalized hermite and laguerre polynomials as well as their higher order and multi index cases they also consider several applications of these operational techniques to the solutions of a wide class of partial differential equations including for example the heat wave and transport equations some of which involve second order laguerre derivatives

differential and integral equations for the 2 iterated appell polynomials in this article a set of differential equations of finite order k th order k n for the 2 iterated appell polynomials are derived particular cases k 1 and k 2 are also considered the integral equations for the appell and 2 iterated appell polynomials are established further as an illustration the differential and integral equations for the 2 iterated generalized bernoulli polynomials are derived this article is first attempt in the direction of deriving integral equations for the appell and 2 iterated appell families and for some members belonging to these families

hermite based appell polynomials properties and applications abstract by employing certain operational methods the authors introduce hermite based appell polynomials some properties of hermite appell polynomials are considered which proved to be useful for the derivation of identities involving these polynomials the possibility of extending this technique to introduce hermite based sheffer polynomials for example hermite laguerre and hermite sister celine s polynomials is also investigated

efficient computations for generalized zernike moments and image recovery abstract zernike moments are a set of orthogonal moments which have been successfully applied in the fields of image processing and pattern recognition an innovative calculation method for zernike moments named generalized zernike moments is presented in this study the generalized zernike moment is a variant of zernike moment in this paper we are proposing methods to calculate high order generalized zernike moments two kinds of recurrence for calculating generalized zernike moments were introduced with rigorous proofs through the usage of the symmetries operated by the dihedral group of order eight the proposed method is fast and stable the experimental results show that of the proposed method took 4 206s to compute the top 500 order generalized zernike moments of an image with 512 by 512 pixels furthermore by choosing the extra parameter in the recurrence the method enhanced the accuracy remarkably compared to the regular zernike moments its normalized mean square error is 0 00144067 when was set to 66 and the top 500 order moments were used to reconstruct the image this error is 40 47 smaller than the one obtained by using the regular zernike moments

non radial solutions for some semilinear elliptic equations on the disk abstract starting with approximate solutions of the equation u w u 3 on the disk with zero boundary conditions we prove that there exist true solutions nearby one of the challenges here lies in the fact that we need simultaneous and accurate control of both the inverse dirichlet laplacean and nonlinearities we achieve this with the aid of a computer using a banach algebra of real analytic functions based on zernike polynomials besides proving existence and symmetry properties we also determine the morse index of the solutions

the role of jacobi polynomials in the theory of hermite and laguerre 2d polynomials using an alternative definition of usual hermite polynomials two problems in the theory of general hermite and laguerre 2d polynomials can be separated with advantage for the further treatment the introduction of a general 2d matrix in the linear transformation of powers of the components of a 2d vector and the generation of hermite or laguerre polynomials by applying an integral operator to these powers the jacobi polynomials appear in the finite dimensional irreducible representations of the two dimensional general linear group gl 2 c

a note on welch s approximate t solution to bioequivalence assessment the crossover is considered to be the design of choice for bioequivalence studies and regulatory authorities currently require evidence of average bioequivalence which is a special case of population bioequivalence to assess average bioequivalence the double t test can be performed without welch s approximation

multi variable hermite matrix polynomials properties and applications abstract in this paper the multi variable hermite matrix polynomials are introduced by algebraic decomposition of exponential operators their properties are established using operational methods the matrix forms of the chebyshev and truncated polynomials of two variable are also introduced which are further used to derive certain operational representations and expansion formulae

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

finding mixed families of special polynomials associated with appell sequences abstract in this paper certain mixed special polynomial families associated with appell sequences are introduced and their properties are established further operational rules providing connections between these families and known special polynomials are established which are used to derive the identities and results for the members of these new families determinantal definitions of the polynomials associated with appell family are also derived the approach presented is general

confidence intervals associated with tests for bioequivalence summary previously the connection between bioequivalence tests and their associated confidence intervals was not well understood in fact the oc level two one sided tests approach was thought to be associated with a 1 20c confidence interval in this paper we build up connections which allow us to construct 1 oc confidence intervals from cx level tests and vice versa when applied to the cc level two one sided tests the resultant 1 oc confidence intervals are properly contained in westlake s 1 oc symmetric interval 1976 our approach is readily generalized to different settings including the nonparametric setting the ratio parameter setting and the repeated confidence interval setting

laguerre 2d functions and their application in quantum optics a set of orthonormalized and complete functions of two real variables in complex representation involving laguerre polynomials as a substantial part is introduced and is referred to as the set of laguerre two dimensional 2d functions the properties of the set of laguerre 2d functions are discussed and the fourier and radon transforms of these functions are calculated the laguerre 2d functions form a basis for a realization of the five dimensional lie algebra to the heisenberg weyl group for a two mode system real representation of this set of functions by a sum over products of hermite functions involving jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between laguerre and hermite polynomials in both directions the set of laguerre 2d functions is the most appropriate set of functions for the fock state representation of quasi probabilities in quantum optics the wigner quasi probability in fock state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding laguerre 2d function the properties of orthonormality and completeness of the laguerre 2d functions provide the fock state matrix elements of the density operator directly from the quasi probabilities the perina mista representation of the glauber sudarshan quasi probability can be represented with advantage by the laguerre 2d functions the fock state matrix elements of the displacement operator and the scalar product of displaced fock states are closely related to laguerre 2d functions

sequential equivalence testing and repeated confidence intervals with applications to normal and binary responses summary we propose group sequential tests of the equivalence of two treatments based on ideas related to repeated confidence intervals these tests adapt readily to unpredictable group sizes to the possibility of continuing even though a boundary has been crossed and to nonnormal observations in comparing two binomial distributions the required sample size depends strongly on the average success probability and an adaptive choice of group size is needed to produce an efficient test meeting specified error probability constraints a special case is the experiment where interim analyses are performed not for the purpose of early termination but simply to adjust the sample size so that nominal error rates will be guaranteed despite the presence of a nuisance parameter

phase space dynamics and hermite polynomials of two variables and two indices the theory of hermite polynomials of two variables and two indices is discussed herein within the context of phase space formulation of classical and quantum mechanics they play the same role as conventional hermite polynomials in ordinary quantum mechanics finally their extension to m variables and m indices is analyzed

optimum allocation of treatments for welch s test in equivalence assessment as an extension of welch s test to equivalence trials in a matched pair design we determine the sample sizes n1 and n2 that maximize the power at given alternatives for a given total sample size n n1 n2 although the optimal allocation is obtained asymptotically when the ratio of the standard deviations in both treatment groups equals the ratio of the sample sizes numerical investigations show that this result does not hold for sample sizes where n or 25 for convenience we provide tables containing the optimal combinations for the bioequivalence problem as well as for the problem of testing a clinically relevant difference results indicate that for small sample sizes the optimum allocations of the treatments differ significantly in both testing problems although they are asymptotically identical in addition we provide simple approximate equations that can be used for the determination of the required sample sizes to control a preassigned probability of type ii error

zernike functions rigged hilbert spaces and potential applications we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing

asymptotic formulae for two variable hermite polynomials new asymptotic formulae for the two dimensional hermite polynomials with large values of indices are found the applications to the photon distribution functions in squeezed one mode mixed quantum states are considered

structural localization phenomena and the dynamical phase space analogy the localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints a from a modal perspective b from a formulation which allows the amplitude to modulate in an asymptotically defined slow space and c from a dynamical analogy in phase space suggested by the form of the underlying differential equation a simple strut on an asymmetric nonlinear foundation provides a typical illustrative example the three approaches emphasize different features of the localization phenomenon the modal view illustrates the distinctive effects of boundary conditions the modulated approach generates a convenient second order differential equation in the amplitude function and the dynamical phase space analogy suggests a useful interpretation of localization as a homoclinic connection comparisons are also made with nonlinear numerical solutions as the strut length approaches infinity it is shown that the fully localized solution represents the unstable post buckled state with the lowest energy allowing evaluation of the minimum energy barrier relevant to dynamical impact studies attention is drawn to the possibility of spatial chaos becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection

comparison of different methods for decision making in bioequivalence assessment if the regulatory requirements are symmetrical the use of symmetrical confidence intervals as a decision rule for bioequivalence assessment leads as shown by simulations to better level properties and an inferior power compared to a rule based on shortest confidence intervals a choice between these two approaches will have to depend on a loss function for asymmetric regulatory requirements symmetrical confidence intervals should not be used however a decision can still be based on posterior probabilities pr theta epsilon r1 r2 x or shortest confidence intervals for purposes of inference presentation and interpretation of results we think that the use of symmetrical confidence intervals alone can be misleading and we therefore recommend that the posterior probabilities and densities or at least the shortest confidence intervals be given

on the use of xfem within the arlequin framework for the simulation of crack propagation abstract the arlequin method is a generic numerical method that allows by local superposition and coupling of models to address multimodel and multiscale mechanical problems in particular this method has already been used to super impose cracked patches on sound structures reducing this way the global simulation resources a classical finite element approach would have required in this paper one of the key features of the extended finite element method namely the heaviside enrichment function is used within the arlequin framework to further reduce the costs of crack propagation simulations the main goal of the paper is to describe the proposed methodology and to assess its performance through numerical experiments

image analysis by generalized chebyshev fourier and generalized pseudo jacobi fourier moments in this paper we present two new sets named the generalized chebyshev fourier radial polynomials and the generalized pseudo jacobi fourier radial polynomials which are orthogonal over the unit circle these generalized radial polynomials are then scaled to define two new types of continuous orthogonal moments which are invariant to rotation the classical chebyshev fourier and pseudo jacobi fourier moments are the particular cases of the proposed moments with parameter 0 the relationships among the proposed two generalized radial polynomials and jacobi polynomials shift jacobi polynomials and the hypergeometric functions are derived in detail and some interesting properties are discussed two recursive methods are developed for computing radial polynomials so that it is possible to improve computation speed and to avoid numerical instability simulation results are provided to validate the proposed moment functions and to compare their performance with previous works highlightspaper presents two generalized radial polynomials that are orthogonal over the unit circle paper constructs two new generalized descriptors using the scaled radial polynomials two recursive strategies for the computation of proposed radial polynomials are presented the proposed polynomials are related to jacobi shift jacobi polynomials and hypergeometric functions the distribution of zeroes of the proposed polynomials can be controlled by free parameter

bridging techniques in a multi scale modeling of pattern formation abstract bridging techniques between microscopic and macroscopic models are discussed in the case of wrinkling analysis the considered macroscopic models are related to envelope equations of ginzburg landau type but generally they are not valid up to the boundary to this end a multi scale approach is considered the reduced model is implemented in the bulk while the full model is applied near the boundary and these two models are coupled with the arlequin method ben dhia 1998 this paper focuses on the definition of the coupling model and the transition between two scales especially a new nonlocal bridging technique is presented and compared with another recent one hu et al 2011 the present method can also be seen as a guide for coupling techniques involving other reduced order models

operational rules and a generalized hermite polynomials in this paper we use operational rules associated with three operators corresponding to a generalized hermite polynomials introduced by szego to derive as far as we know new proofs of some known properties as well as new expansions formulae related to these polynomials

operational methods and laguerre gould hopper polynomials abstract in this paper the authors introduce the laguerre gould hopper polynomials by combining the operational methods with the principle of monomiality generating function series definition differential equation and certain other properties of laguerre gould hopper polynomials are derived further operational representations of these polynomials are established which are used to get integral representations and expansion formulae for these polynomials

levy stable two sided distributions exact and explicit densities for asymmetric case we investigate functions g ensuremath alpha ensuremath beta x which are heavy tailed l evy stable probability distributions of index 0l ensuremath alpha ensuremath leqslant 2 and appropriate asymmetry parameter ensuremath beta they are of central importance in physics of amorphous and disordered systems econophysics geology hydrology internet traffic dynamics of human relations etc we present an ensemble of exact and explicit solutions of g ensuremath alpha ensuremath beta x for all admissible rational values of ensuremath alpha and ensuremath beta we reproduce all the previously known cases and furnish distributions for values ensuremath alpha and ensuremath beta which were previously inaccessible we point out many instances of experimental and statistical data that could be described by our results

some connection and linearization problems for polynomials in and beyond the askey scheme the connection problem is considered in a hypergeometric function framework for i the two most general families of polynomials belonging to the askey scheme wilson and racah and ii some generalized laguerre and jacobi polynomials falling outside that scheme sister celine cohen and prabhakar jain which are relevant to the study of quantum mechanical systems and include as particular cases the generalizations of the classical families with sobolev type orthogonality in addition using the same method three new linearization like formulae for the gegenbauer polynomials are also derived a linearization formula that generalizes the m n case of dougall s formula the analogue of the m n case of nielsen s inverse linearization formula for hermite polynomials and a connection formula for the squares closed analytical formulae for the corresponding connection and linearization coe cients are given in terms of hypergeometric functions of unit argument which at times can be further simpli ed and expressed as single hypergeometric terms c 2001 published by elsevier science b v

differentiability of bizonal positive definite kernels on complex spheres abstract we prove that any continuous function with domain z c z 1 that generates a bizonal positive definite kernel on the unit sphere in c q q 3 is continuously differentiable in z c z 1 up to order q 2 with respect to both z and z in particular the partial derivatives of the function with respect to x re z and y im z exist and are continuous in z c z 1 up to the same order

recursive computation of generalised zernike polynomials abstract an algorithmic approach for generating generalised zernike polynomials by differential operators and connection matrices is proposed this is done by introducing a new order of generalised zernike polynomials such that it collects all the polynomials of the same total degree in a column vector the connection matrices between these column vectors composed by the generalised zernike polynomials and a family of polynomials generated by a rodrigues formula are given explicitly this yields a rodrigues type formula for the generalised zernike polynomials themselves with properly defined differential operators another consequence of our approach is the fact that the generalised zernike polynomials obey a rather simple partial differential equation we recall also how to define hermite zernike polynomials

product of two hypergeometric functions with power arguments abstract the hypergeometric product formula describes the product of two hypergeometric functions with arguments xt and yt respectively the result is a power series in t whose coefficients are hypergeometric polynomials brychkov has extended the product formula such that in one argument the variable t can be squared we obtain a further extension such that in both arguments the variable t can be raised to positive integer powers

a parallel computer implementation of the asymptotic numerical method to study thermal convection instabilities we have developed a numerical model to efficiently compute steady state combined buoyancy and thermocapillary convection solutions it features a parallel computer implementation of an asymptotic numerical method to perform steady state path following and locate bifurcation points in problems involving large size algebraic systems up to few million degrees of freedom the model has been first validated on a problem for which a reference solution exists and then is used to analyse the influence of the container size and shape on rayleigh benard marangoni convection and its related cellular pattern

influence of localized imperfections on the buckling of cylindrical shells under axial compression the influence of localized imperfections on the buckling of a long cylindrical shell under axial compression is analysed by using a double scale analysis including interaction modes this leads to a system of coupled complex non linear differential equations with discontinuous derivatives we propose analytical formulas to predict the reduction of the critical buckling load

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

an extension of welch s approximate t solution to comparative bioequivalence trials summary we consider the problem of proving statistically the equivalence of two treatments with normally distributed observations the size and the power of the commonly used test procedures which assume equality of the variances are investigated when this assumption is violated on the one hand the preassigned level is considerably exceeded and on the other hand the power if no treatment difference exists drops heavily for differing sample sizes we propose an extension of welch s approximate t solution for the behrens fisher problem in bioequivalence assessment which is asymptotically optimal in a certain subclass of tests the behaviour of the proposed test is investigated in various finite sample situations the results show that the extension of welch s approximate t solution should be preferred for testing bioequivalence of two treatments whenever the population variances cannot be assumed equal and the sample sizes are different sometimes the experimenter knows which sample has the larger variance in this case the larger sample size should always be assigned to that group with the larger variance in order to optimize the actual level and power of the test

the higher order heat type equations via signed levy stable and generalized airy functions we study the higher order heat type equation with first time and mth spatial partial derivatives m 2 3 we demonstrate that its exact solutions for m even can be constructed with the help of signed levy stable functions for m odd the same role is played by a special generalization of the airy ai function that we introduce and study this permits one to generate the exact and explicit heat kernels pertaining to these equations we examine analytically and graphically the spatial and temporary evolution of particular solutions for simple initial conditions

fractional exponential operators and nonlinear partial fractional differential equations in the weyl fractional derivatives in this article we introduce fractional exponential operator and give an integral representation for it moreover we obtain the formal solutions of some nonlinear partial fractional differential equations using this operator

a semi analytical buckling analysis of imperfect cylindrical shells under axial compression abstract in the framework of the cellular bifurcation theory we investigate the effect of distributed and or localized imperfections on the buckling of long cylindrical shells under axial compression using a double scale perturbative approach including modes interaction we establish that the evolution of amplitudes of instability patterns is governed by a non homogeneous second order system of three non linear complex equations the localized imperfections are included by employing jump conditions for their amplitude and permitting discontinuous derivatives by solving these amplitude equations we show the influence of distributed and or localized imperfections on the reduction of the critical load to assess the validity of the present method our results are compared to those given by two finite element codes

taylor meshless method for solving non linear partial differential equations a true meshless integration free method based on taylor series named taylor meshless method tmm has been proposed recently to solve partial differential equations pdes where the shape functions are high degree polynomials and the discretization concerns only the boundary with high computational efficiency and exponential convergence the tmm has been confirmed to be very robust in linear problems including large scale cases in this paper the tmm and the automatic differentiation ad are combined with the newton method to solve non linear elliptic pdes where the ad is used to compute shape functions in a fast manner the numerical results illustrate that the proposed algorithm is very accurate and efficient in solving non linear elliptic problems

multivariable lagrange expansion and generalization of carlitz srivastava mixed generating functions abstract families of mixed generating functions generalizing those of the carlitz srivastava type are derived here by applying methods based on the multivariable extension of the lagrange expansion it is also shown that the combination with techniques of operational nature offers a wide flexibility to explore a wealth of mixed bilateral generating functions for special functions with many variables

spherical geometry zernike s separability and interbasis expansion coefficients free motion on a 3 sphere properly projected on the 2 dimensional manifold of a disk yields the zernike system which exhibits the fundamental properties of superintegrability these include separability in a variety of coordinate systems polynomial solutions and a particular subset of clebsch gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the askey scheme deriving these results from the initial formulation in spherical geometry provides the zernike system with interest beyond its optical applications

a new fourier related double scale analysis for instability phenomena in sandwich structures abstract in this paper we present a new fourier related double scale analysis to study instability phenomena of sandwich structures by using the technique of slowly variable fourier coefficients a zig zag theory based microscopical sandwich model is transformed into a macroscopical one that offers three numerical advantages firstly only the envelopes of instability patterns are evaluated and this leads to a significant improvement on computational efficiency especially when dealing with high wavenumber wrinkling phenomena secondly the proposed macroscopical model allows one to select modal wavelength which makes easy to control non linear calculations thirdly in contrast to landau ginzburg envelope equations it may also remain valid away from the bifurcation point and the coupling between global and local instabilities can be accounted for the established non linear system is solved by asymptotic numerical method anm which is more reliable and less time consuming than other iterative classical methods the proposed double scale analysis yields accurate results with a significant reduced computational cost

transformations of laguerre 2d polynomials with applications to quasiprobabilities laguerre 2d polynomials are defined and their properties are investigated the laguerre 2d functions introduced in 1 2 are related to the laguerre 2d polynomials in such a way that they also include the weight function for the orthonormalization of the laguerre 2d polynomials a one parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied in particular to hermite polynomials and to laguerre 2d polynomials these transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments they contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for hermite and laguerre 2d polynomials applications of the laguerre 2d polynomials and 2d functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the fock state basis and by normally and otherwise ordered moments are considered the inversion of these representations is obtained in all cases a restricted design of quasiprobabilities should become possible

generalized polynomials operational identities and their applications abstract it is shown that an appropriate combination of methods relevant to generalized operational calculus and to special functions can be a very useful tool to treat a large body of problems both in physics and mathematics we discuss operational methods associated with multivariable hermite laguerre legendre and other polynomials to derive a wealth of identities useful in quantum mechanics electromagnetism optics etc or to derive new identities between special functions as e g mehler or mixed type generating functions

polyharmonic bergman spaces and bargmann type transforms abstract in the main result of the paper we prove the decomposition of polyharmonic bergman spaces over the upper half plane into spaces of polyanalytic functions then we introduce the decomposition of polyharmonic bergman spaces into the orthogonal sum of its true polyharmonic bergman subspaces and we state isometric isomorphisms between the different true polyharmonic bergman spaces this allows us to define the k th harmonic hilbert component of a polyharmonic bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic bergman spaces the harmonic complex fourier transform is introduced in order to give an explicit description of the cartesian and the laguerre harmonic components of the images of a bargmann type transform for the true polyharmonic bergman spaces finally it is proved that the polyharmonic bergman space of order j is isometric isomorphic to 2j copies of the corresponding hardy space

some properties of hermite based sheffer polynomials abstract in this paper the concepts and the formalism associated with monomiality principle and sheffer sequences are used to introduce family of hermite based sheffer polynomials some properties of hermite sheffer polynomials are considered further an operational formalism providing a correspondence between sheffer and hermite sheffer polynomials is developed furthermore this correspondence is used to derive several new identities and results for members of hermite sheffer family

operational identities and properties of ordinary and generalized special functions the theory of hermite laguerre and of the associated generating functions is reformulated within the framework of an operational formalism this point of view provides more efficient tools which allow the straightforward derivation of a wealth of new and old identities in this paper a central role is played by negative derivative operators and by their link with the tricomi functions and the generalized laguerre polynomials

generalized zernike or disc polynomials we investigate generalized zernike or disc polynomials p m n z z which are orthogonal 2d polynomials in the unit disc 0 zz 1 is a free parameter these polynomials can be expressed by jacobi polynomials of transformed arguments in connection with a simple angle dependence a limiting procedure leads to laguerre 2d polynomials lm n z z furthermore we introduce the corresponding orthonormalized disc functions the disc polynomials and disc functions obey two differential equations a first order and a second order one with a certain degree of freedom and the operators of lowering and raising of the indices are found these operators can be closed to a lie algebra su 1 1 su 1 1 new generating functions are derived from an operational representation which is alternative to the rodrigues type representation the one dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0 r 1 with weight factors 1 r2 x are ultraspherical or gegenbauer polynomials in a new standardization the lowering and raising operators to the corresponding orthonormalized functions form a simple su 1 1 lie algebra this is given in the appendix in sketched form

corrigendum general hermite and laguerre two dimensional polynomials by comparing formula 5 5 with formula a 4 in appendix a a difference in the order of indices v xy in 5 5 with the corresponding v 21 in a 4 was overlooked this has an influence on the orthogonality and completeness relations of the hermite 2d functions which possess the following correct form with an additional transposition notation of u 1 and analogously for the laguerre 2d functions by substitution where is the contragredient matrix to u in order to get full compatibility with appendix a one may substitute in section 5 matrix v by its transposed v i e v v vxy vyx and vzz vz z

some properties of legendre gould hopper polynomials and operational methods abstract in this paper the author introduces the legendre gould hopper polynomials by combining the operational methods with the principle of monomiality generating functions series definition differential equation and certain other properties of legendre gould hopper polynomials are derived further operational representations of these polynomials are established which are used to get integral representations and expansion formulae certain summation formulae for these polynomials are also obtained

levy flights exact results and asymptotics beyond all orders a comprehensive study of the symmetric levy stable probability density function is presented this is performed for orders both less than 2 and greater than 2 the latter class of functions are traditionally neglected because of a failure to satisfy non negativity the complete asymptotic expansions of the symmetric levy stable densities of order greater than 2 are constructed and shown to exhibit intricate series of transcendentally small terms asymptotics beyond all orders it is demonstrated that the symmetric levy stable densities of any arbitrary rational order can be written in terms of generalized hypergeometric functions and a number of new special cases are given representations in terms of special functions a link is shown between the symmetric levy stable density of order 4 and pearcey s integral which is used widely in problems of optical diffraction and wave propagation this suggests the existence of applications for the symmetric levy stable densities of order greater than 2 despite their failure to define a probability density function

new relations for two dimensional hermite polynomials the effective formulas reducing the two dimensional hermite polynomials to the classical one dimensional orthogonal polynomials by jacobi gegenbauer legendre laguerre and hermite are given new one parameter generating functions for the diagonal multidimensional hermite polynomials are derived the factorial moments and cumulants of the distribution functions related to the hermite polynomials of two variables with equal indices are expressed in terms of the legendre and chebyshev polynomials asymptotical formulas for the two dimensional polynomials with large values of indices and zero arguments are found the applications to the squeezed one mode states and to the time dependent quantum harmonic oscillator are considered

a probabilistic approach to some results by nieto and truax it is shown that the usual operational methods applied to characterize the properties of gould and hopper polynomials as well as the properties of quantum higher order and squeezed states can be replaced by a probabilistic approach using stable distributions

a bridging technique to analyze the influence of boundary conditions on instability patterns in this paper we present a new numerical technique that permits to analyse the effect of boundary conditions on the appearance of instability patterns envelope equations of landau ginzburg type are classically used to predict pattern formation but it is not easy to associate boundary conditions for these macroscopic models indeed envelope equations ignore boundary layers that can be important for instance in cases where the instability starts first near the boundary in this work the full model is considered close to the boundary an envelope equation in the core and they are bridged by the arlequin method 1 simulation results are presented for the problem of buckling of long beams lying on a non linear elastic foundation

the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

generalized shift operators and pseudo polynomials of fractional order we introduce families of pseudo kampe de feriet polynomials which can be viewed as the natural complement for the theory of fractional derivatives and partial fractional differential equations of evolutive type we show that they allow the possibility of treating a large variety of exponential operators providing generalized fractional forms of shift operators the link of these pseudo polynomials with wright type functions is also discussed

on fractional calculus of a2n 1 x function abstract in this article we consider the a 2 n 1 x function the generalized airy function a 2 n 1 x 0 cos zx 1 n 1 z 2 n 1 2 n 1 dz and find some properties of this function and its derivatives these properties are included in integral representations for product of a 2 n 1 x function a 2 n 1 x wavelets and integral and series identities moreover the a 2 n 1 x polynomials are derived and bi orthogonality properties with their associated functions are obtained

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other