characterization of the dunkl classical symmetric orthogonal polynomials in this paper we introduce the notion of dunkl classical orthogonal polynomials then we show that generalized hermite and generalized gegenbauer polynomials are the only dunkl classical symmetric orthogonal polynomials by solving a suitable differential difference equation

on the coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

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

connection coefficients between boas buck polynomial sets abstract in this paper a general method to express explicitly connection coefficients between two boas buck polynomial sets is presented as application we consider some generalized hypergeometric polynomials from which we derive some well known results including duplication and inversion formulas

asymptotics of orthogonal polynomial s entropy this is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree this study is motivated primarily by quantum mechanics where the wave functions and the densities of the states of solvable quantum mechanical systems are expressed by means of orthogonal polynomials moreover the uncertainty principle lying in the ground of quantum mechanics is best formulated by means of position and momentum entropies in this sense the behavior for large values of the degree is intimately connected with the information characteristics of high energy states but the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials we describe some results obtained in the last 15 years as well as sketch the ideas behind their proofs

linearization and krein like functionals of hypergeometric orthogonal polynomials the krein like r functionals of the hypergeometric orthogonal polynomials pn x with the kernel of the form xs x pm1 x pmr x being x the weight function on the interval r are determined by means of the srivastava linearization method the particular 2 functionals which are particularly relevant in quantum physics are explicitly given in terms of the degrees and the characteristic parameters of the polynomials they include the well known power moments and the novel krein like moments moreover various related types of exponential and logarithmic functionals are also investigated

fourth order difference equation for the first associated of classical discrete orthogonal polynomials we derive the fourth order difference equation satisfied by the first associated of classical orthogonal polynomials of a discrete variable we give it explicitly for first associated of hahn polynomials from which can be derived by a limiting process the equation satisfied by first associated of all classical families continuous and discrete

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

the moduli space of cubic fourfolds via the period map we characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space it follows then that the geometric invariant theory git compactification of the moduli space of cubic fourfolds is isomorphic to the looijenga compactification associated to this arrangement this paper builds on and is a natural continuation of our previous work on the git compactification of the moduli space of cubic fourfolds

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

renyi entropies lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions the quantification of the spreading of the orthogonal polynomials p n x can be investigated by means of the renyi entropies r q r q being a positive integer number of the associated rakhmanov probability densities r x email protected x p n 2 x where w x is the corresponding weight function the renyi entropies are closely related to the l q norms of the polynomials in this manuscript the l q norms and the associated renyi entropies of the real hypergeometric orthogonal polynomials i e hermite laguerre and jacobi polynomials and the generalized hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of lauricella and srivastava daoust types which are evaluated at some specific values of 2q variables these functions depend on 4q 1 and 6q 2 parameters respectively which are determined by the order q the degree n of the polynomial and the parameters of the orthogonality weight function w x the key idea is based on some extended linearization formulas for these polynomials these results open the way to determine the renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials

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

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable i e associated with the names of charlier meixner krawtchouk and hahn also a recurrence relation is given for the coefficients in the formula expressing the nth associated polynomial in terms of the original polynomials

cluster splitting bifurcation in a system of coupled maps abstract we consider cluster splitting bifurcations in a system of globally coupled maps as coupling parameter decreases at these transitions the number of clusters i e groups of elements with identical dynamics increases we demonstrate that different cascades of cluster splitting can occur depending on statistics of redistribution of the oscillators between new born clusters

using advection to control the velocity of patterns in rings of maps traveling patterns are well known features of rings of symmetrically coupled maps which however propagate with rather small velocities of the order of 10 3 sites per step we show that it is easy to produce traveling patterns with velocities tunable over three orders of magnitude by simply breaking the symmetry of the coupling between neighbors this asymmetry arises naturally when the usual model of coupled map lattices is generalized to also include advection in addition asymmetries change the wavelength of waves traveling on the lattice 2002 elsevier science b v all rights reserved

connection problems via lowering operators the lowering operator associated with a polynomial set pn n 0 is an operator not depending on n and satisfying the relation pn npn 1 in this paper we express explicitly the connection coefficients between two polynomial sets using their corresponding lowering operators we obtain some well known results as particular cases including some duplication and addition formulas

connection coefficients for laguerre sobolev orthogonal polynomials laguerre sobolev polynomials are orthogonal with respect to an inner product of the form p qs 0 p x q x x e x dx 0 p x q x d x w here 1 0 and p q p the linear space of polynomials with real coefficients if d x x e x dx the corresponding sequence of monic orthogonal polynomials q x has been studied by marcellan et al j com put appl math 71 1996 245 265 while if d x x dx the sequence of monic orthog onal polynomials l x was introduced by koekoek and meijer siam j math anal 24 1993 768 782 for each of these two families of laguerre sobolev polynomials here we give the explicit expression of the connection coefficients in their expansion as a series of standard la guerre polynomials the inverse connection problem of expanding laguerre polynomials in series of laguerre sobolev polynomials and the connection problem relating two families of laguerre sobolev polynomials with different parameters are also considered

expansion of multivariable polynomials in products of orthogonal polynomials in one variable we propose an approach to develop multivariable polynomials in multiple series of orthogonal polynomials in one variable the action of a partial differential operator on a series of products of classical orthogonal polynomials is first analyzed and permits the generation of recurrence relations for the expansion coefficients like in the one variable case polynomial solutions of linear partial differential equations pdes with polynomial coefficients are also examined giving new results on harmonic polynomials based on the appell property of hermite polynomials

classical orthogonal polynomials dependence on parameters most of the classical orthogonal polynomials continuous discrete and their q analogues can be considered as functions of several parameters ci a systematic study of the variation innitesimal and nite of these polynomials pn x ci with respect to the parameters ci is proposed a method to get recurrence relations for connection coecients linking r c r i pn x ci t opn x ci is given and in some situations explicit expressions are obtained this allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function a basic theorem on the zeros of ci pn x ci is also proved c 2000 elsevier science b v all rights reserved msc 33c25 42c05 33b15

algorithms for the integration and derivation of chebyshev series general formulas for the mth integral and derivative of a chebyshev polynomial of the first or second kind are presented the result is expressed as a finite series of the same kind of chebyshev polynomials these formulas permit to accelerate the determination of such integrals or derivatives besides it is presented formulas for the mth integral and derivative of finite chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

modified clebsch gordan type expansions for products of discrete hypergeometric polynomials abstract starting from the second order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials p n we find the analytical expressions of the expansion coefficients of any polynomial rm x and of the product rm x qj x in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric polynomials here qj x denotes an arbitrary discrete hypergeometric polynomial of degree j the particular cases in which r m corresponds to the non orthogonal families x m the rising factorials or pochhammer polynomials x m and the falling factorial or stirling polynomials x m are considered in detail the connection problem between discrete hypergeometric polynomials which here corresponds to the product case with m 0 is also studied and its complete solution for all the classical discrete orthogonal hypergeometric cdoh polynomials is given also the inversion problems of cdoh polynomials associated to the three aforementioned nonorthogonal families are solved

on the connection coefficients and recurrence relations arising from expansions in series of laguerre polynomials a formula expressing the laguerre coefficients of a general order derivative of an infinitely differentiable function in terms of its original coefficients is proved and a formula expressing explicitly the derivatives of laguerre polynomials of any degree and for any order as a linear combination of suitable laguerre polynomials is deduced a formula for the laguerre coefficients of the moments of one single laguerre polynomial of certain degree is given formulae for the laguerre coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its laguerre coefficients are also obtained a simple approach in order to build and solve recursively for the connection coefficients between jacobi laguerre and hermite laguerre polynomials is described an explicit formula for these coefficients between jacobi and laguerre polynomials is given of which the ultra spherical polynomials of the first and second kinds and legendre polynomials are important special cases an analytical formula for the connection coefficients between hermite and laguerre polynomials is also obtained

the spectral connection matrix for classical orthogonal polynomials of a single parameter abstract in this paper we study the so called connection problem of given a polynomial expressed in the basis of one set of orthogonal polynomials computing the coefficients with respect to a different set of orthogonal polynomials we restrict our current study to the classical real orthogonal polynomials of the hermite laguerre and gegenbauer including legendre families the computational tool for this work is the class of quasiseparable matrices while the relationships between orthogonal polynomials and rank structured matrices such as quasiseparable matrices are very well known in this paper we investigate a more recently considered relationship we prove that while the matrix that implements the desired connection is not itself quasiseparable it is an eigenvector matrix of one that is quasiseparable we suggest to refer to this structured matrix as the spectral connection matrix finally we present a simple algorithm exploiting the computationally favorable properties of quasiseparable matrices to implement the desired change of basis by exploiting the quasiseparable structure this algorithm enjoys an order of magnitude reduction of complexity as compared to the simple method of inverting the connection matrix directly while not the focus of the paper some very preliminary numerical experimentation shows some positive indications that even with this reduction in complexity the accuracy of the resulting change of basis algorithm is comparable to that of inverting the connection matrix directly

hermite sobolev and closely connected orthogonal polynomials abstract in this paper a bilinear functional a g m o n m c m m g m which corresponds to the hermite sobolev and closely connected orthogonal polynomials is studied the domain of m m 1 n in which a is positive definite is given and the hermite sobolev and closely connected formal orthogonal polynomials are given in the nonnormal case

on the numerical evaluation of linear recurrences we present some remarks on the numerical evaluation of recurrence relations rounding error bounds are presented of the numerical scheme and some numerical examples are given in particular we analyse conversion recurrences from different families of orthogonal polynomials the limit case of jacobi sobolev polynomials random recurrences and perturbed gegenbauer polynomials in all these examples the theoretical bounds give sharp relative rounding error estimations the parallel evaluation of recurrences are also considered and numerical tests on a cray t3d are presented

one dimensional dynamics for traveling fronts in coupled map lattices multistable coupled map lattices typically support traveling fronts separating two adjacent stable phases we show how the existence of an invariant function describing the front profile allows a reduction of the infinitely dimensional dynamics to a one dimensional circle homeomorphism whose rotation number gives the propagation velocity the mode locking of the velocity with respect to the system parameters then typically follows we study the behavior of fronts near the boundary of parametric stability and we explain how the mode locking tends to disappear as we approach the continuum limit of an infinite density of sites

lyapunov spectra of periodic orbits for a many particle system the lyapunov spectrum corresponding to a periodic orbit for a two dimensional many particle system with hard core interactions is discussed noting that the matrix to describe the tangent space dynamics has the block cyclic structure the calculation of the lyapunov spectrum is attributed to the eigenvalue problem of 16 16 reduced matrices regardless of the number of particles we show that there is the thermodynamic limit of the lyapunov spectrum in this periodic orbit the lyapunov spectrum has a step structure which is explained by using symmetries of the reduced matrices

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

on the limit relations between classical continuous and discrete orthogonal polynomials abstract limit relations between classical continuous jacobi laguerre hermite and discrete charlier meixner kravchuk hahn orthogonal polynomials are well known and can be described by relations of type lim pn x qn x deeper information on these limiting processes can be obtained from the expansion p n x k 0 r k x n k in this paper a method for the recursive computation of coefficients rk x n is designed being the main tool the consideration of a closely related connection problem which can be solved also recurrently by using an algorithm recently developed by the authors

connection problems for polynomial solutions of nonhomogeneous differential and difference equations abstract we consider nonhomogeneous hypergeometric type differential difference and q difference equations whose nonhomogeneity is a polynomial q n x the polynomial solution of these problems is expanded in the q n x basis and also in a basis p n x related in a natural way with the homogeneous hypergeometric equation we give an algorithm building a recurrence relation for the expansion coefficients in both bases that we solve explicitly in many cases involving classical orthogonal polynomials finally some concrete applications and extensions are given

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

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

stable synchronised states of coupled tchebyscheff maps abstract coupled tchebyscheff maps have recently been introduced to explain parameters in the standard model of particle physics using the stochastic quantisation of parisi and wu this paper studies dynamical properties of these maps finding analytic expressions for a number of periodic states and determining their linear stability numerical evidence is given for nonlinear stability of some of these states and also the presence of exponentially slow dynamics for some ranges of the parameter these results indicate that a theory of particle physics based on coupled map lattices must specify strong physical arguments for any choice of initial conditions and explain how stochastic quantisation is obtained in the many stable parameter regions

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

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 theory of wavelet transform method on chaotic synchronization of coupled map lattices the wavelet transform method originated by wei et al phys rev lett 89 284103 4 2002 was proved juang and li j math phys 47 072704 16 2006 juang et al j math phys 47 122702 11 2006 shieh et al j math phys 47 082701 10 2006 to be an effective tool to reduce the order of coupling strength for coupled chaotic systems to acquire the synchrony regardless the size of oscillators in juang et al ieee trans circuits syst i regul pap 56 840 2009 such method was applied to coupled map lattices cmls it was demonstrated that by adjusting the wavelet constant of the method can greatly increase the applicable range of coupling strengths the parameters range of the individual oscillator and the number of nodes for local synchronization of cmls no analytical proof is given there in this paper the optimal or near optimal wavelet constant can be explicitly identified as a result the above described scenario can be rigorously verified

comments on monomiality ordinary polynomials and associated bi orthogonal functions we use laguerre and hermite polynomials to show that the monomiality principle can be exploited to study the properties of the polynomials and of the associated biorthogonal functions

on the connection and linearization problem for discrete hypergeometric q polynomials abstract in the present paper starting from the second order difference hypergeometric equation on the non uniform lattice x s satisfied by the set of discrete hypergeometric orthogonal q polynomials p n we find analytical expressions of the expansion coefficients of any q polynomial r m x s on x s and of the product r m x s q j x s in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric q polynomials

general sobolev orthogonal polynomials abstract in this paper we study orthogonal polynomials with respect to the inner product f g s n u fg m 1 n m u f m g m where m 0 for m 1 n and u is a semiclassical positive definite linear functional for these non standard orthogonal polynomials algebraic and differential properties are obtained as well as their representation in terms of the standard orthogonal polynomials associated with u

some clebsch gordan type linearisation relations and other polynomial expansions associated with a class of generalised multiple hypergeometric series arising in physical and quantum chemical applications in this sequel to srivastava s work 1987 a number of new expansions in series of various classes of hypergeometric polynomials are derived for the general multivariable hypergeometric function which provides an interesting and useful unification of numerous families of hypergeometric functions of one two or more variables encountered naturally in a wide variety of physical and quantum chemical applications by suitably specialising some of these polynomial expansions clebsch gordan type linearisation relations are deduced for the products of several jacobi or laguerre polynomials it is also shown how one of the linearisation relations involving laguerre polynomials would yield the corrected and modified versions of a couple of results given by niukkanen 1985

rational intersection cohomology of quotient varieties given a linear action of a complex reductive group g on a complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory 18 from the construction of this quotient it is not immediately obvious how its topology is related to the topology of x and the group action however when x is nonsingular it is possible at least if one technical condition is satisfied to compute the intersection betti numbers of x g in terms of the rational cohomology of certain nonsingular linear sections z of x and the classifying spaces of certain reductive subgroups h of g see 15 the technical condition is that there should exist at least one point of x which is properly stable for the action of g this is satisfied in most interesting examples such as those coming from moduli problems the sections of x and subgroups of g which occur have a simple description in terms of the representation of g which induces the linear action on x it thus seems reasonable to ask what happens when x is singular the aim of this paper is to show that the procedure for computing the intersection betti numbers of x g can be generalised to the singular case see 2 25 and 2 28 however now one needs to know not only the intersection cohomology of the various sections z of x but also how they are positioned relative to the singularities of x in the special case when they meet the singularities transversely then one has exactly the same formulas as when x is nonsingular except that ordinary cohomology is replaced by intersection cohomology throughout the situation which it is most important to understand is that in which every semistable point is stable and this case is considered in detail in w 2 after the necessary preliminaries have been covered in w 1 some brief remarks are made about the more general case when one only assumes that there exist some stable points a simple example is worked out 2 27 which shows that the formulas can also be used in reverse to compute the intersection betti numbers of x from knowledge of the intersection betti numbers of the quotient x g even simply from the knowledge that these vanish in dimensions greater than dim x g

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

bivariate krawtchouk polynomials in this paper we present a recurrent procedure to solve an inversion problem for monic bivariate krawtchouk polynomials written in vector column form giving its solution explicitly as a by product a general connection problem between two vector column of monic bivariate krawtchouk families is also explicitly solved moreover in the non monic case and also for krawtchouk families several expansion formulas are given but for polynomials written in scalar form

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

transverse limits in the askey tableau limit relations between classical discrete charlier meixner kravchuk hahn to classical continuous jacobi laguerre hcrmite orthogonal polynomials to be called transverse limits are nicely presented in the askey tableau and can be described by relations of type lim 0 p n x s q n s where p n s and q n s stand for the discrete and continuous family respectively deeper information on these limiting processes can be obtained by expanding the discrete polynomial family as p n x s y k r k s n in this paper a method for computing the coefficients r k s n is designed the main tool being the consideration of some closely related connection and inversion problems

some results on quasi monomiality a polynomial set p n n 0 is called quasi monomial if and only if it is possible to define two operators p and m independent of n such thatp p n x np n 1 x andm p n x p n 1 x in this paper we show that every polynomial set is quasi monomial and we present some useful tools to explicitly express the p and m operators for some polynomial families given by their generating functions the obtained results are applied to boas buck polynomial sets

a basic class of symmetric orthogonal polynomials of a discrete variable abstract by using a generalization of sturm liouville problems in discrete spaces a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters which generalizes all classical discrete symmetric orthogonal polynomials is introduced the standard properties of these polynomials such as a second order difference equation an explicit form for the polynomials a three term recurrence relation and an orthogonality relation are presented it is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class moreover moments corresponding to these weight functions can be explicitly computed finally a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail

the combinatorics of a class of representation functions abstract in the study of the irreducible representations of the unitary group u n one encounters a class of polynomials defined on n 2 indeterminates z ij 1 i j n which may be arranged into an n n matrix array z z ij these polynomials are indexed by double gelfand patterns or equivalently by pairs of column strict young tableaux of the same shape using the double labeling property one may define a square matrix d z whose elements are the double indexed polynomials these matrices possess the remarkable group multiplication property d xy d x d y for arbitrary matrices x and y even though these matrices may be singular for z u u n these matrices give irreducible unitary representations of u n these results are known but not always fully proved from the extensive physics literature on representation of the unitary groups where they are often formulated in terms of the boson calculus and the multiplication property is unrecognized the generality of the multiplication property is the key to understanding group representation theory from the purview of combinatorics the combinatorial structure of the general polynomials is expected to be intricate and in this paper we take the first step to explore the combinatorial aspects of a special class which can be defined in terms of the set of integral matrices with given row and column sums these special polynomials are denoted by l z where and are integral vectors representing the row sums and column sums of a class of integral matrices we present a combinatorial interpretation of the multiplicative properties of these polynomials we also point out the connections with macmahon s master theorem and schwinger s inner product formula which is essentially equivalent to macmahon s master theorem finally we give a formula for the double pfaffian which is crucial in the studies of the generating function of the 3 n j coefficients in angular momentum theory we also review the background of the general polynomials and give some of their properties

results for some inversion problems for classical continuous and discrete orthogonal polynomials explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials charlier meixner krawtchouck hahn hahn eberlein into the falling factorial basis are given the corresponding inversion problems are solved explicitly this is done by using a general algorithm recently developed by the authors which is also applied to this kind of inversion problem but relating the basis and the classical continuous orthogonal polynomials of jacobi laguerre hermite and bessel

on linearization and connection coefficients for generalized hermite polynomials abstract we consider the problem of finding explicit formulas recurrence relations and sign properties for both connection and linearization coefficients for generalized hermite polynomials most of the computations are carried out by the computer algebra system maple using appropriate algorithms

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

on the linearization problem involving pochhammer symbols and their q analogues abstract in this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q polynomials in the exponential lattice x s c1qs c3 some simple examples are worked out in detail

recurrences and explicit formulae for the expansion and connection coefficients in series of bessel polynomials a formula expressing explicitly the derivatives of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another explicit formula which expresses the bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original bessel coefficients is also given a formula for the bessel coefficients of the moments of one single bessel polynomial of certain degree is proved a formula for the bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between bessel bessel polynomials is described an explicit formula for these coefficients between jacobi and bessel polynomials is given of which the ultraspherical polynomial and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre bessel and hermite bessel are also developed

mode locking in coupled map lattices abstract we study propagation of pulses along one way coupled map lattices which originate from the transition between two superstable states of the local map the velocity of the pulses exhibits a staircase like behaviour as the coupling parameter is varied for a piecewise linear local map we prove that the velocity of the wave has a devil s staircase dependence on the coupling parameter a wave travelling with rational velocity is found to be stable to parametric perturbations in a manner akin to rational mode locking for circle maps we provide evidence that mode locking is also present for a broader range of maps and couplings

recurrence relations for connection coefficients between q orthogonal polynomials of discrete variables in the non uniform lattice x s q2s we obtain the structure relations for q orthogonal polynomials in the exponential lattice and from these we construct the recurrence relation for the connection coefficients between two families of polynomials belonging to the classical class of discrete q orthogonal polynomials an explicit example is also given

continuous symmetrized sobolev inner products of order n i abstract given a symmetrized sobolev inner product of order n the corresponding sequence of monic orthogonal polynomials q n satisfies that q 2 n x p n x 2 q 2 n 1 x x r n x 2 for certain sequences of monic polynomials p n and r n in this paper we deduce the integral representation of the inner products such that p n and r n are the corresponding sequences of orthogonal polynomials moreover we state a relation between both inner products which extends the classical result for symmetric linear functionals

differential equation of appell polynomials via the factorization method let pn x n 0 be a sequence of polynomials of degree n we define two sequences of differential operators n and n satisfying the following properties n pn x pn 1 x n pn x pn 1 x by constructing these two operators for appell polynomials we determine their differential equations via the factorization method introduced by infeld and hull rev mod phys 23 1951 21 the differential equations for both bernoulli and euler polynomials are given as special cases of the appell polynomials

general linearization formulae for products of continuous hypergeometric type polynomials the linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials hps of a continuous variable which consists of the expansion of the product of two arbitrary hps in series of an orthogonal hp set here this problem is algebraically solved directly in terms of the coefficients of the second order differential equations satisfied by the involved polynomials general expressions for the expansion coefficients are given in integral form and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis hermite laguerre and jacobi as well as several generalized linearization formulae involving these families the connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions which often reduce to a single function of the same type when possible these functions are evaluated in closed form in some cases sign properties of the coefficients such as positivity or non negativity conditions are derived as a by product from their resulting explicit representations

generalized linearization problems we prove that if pi x and pj x are two families of semi classical orthogonal polynomials all the linearization coefficients li j k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index this property also extends to the linearization coefficients arising from an arbitrary number of products of semi classical orthogonal polynomials

orthogonality of some polynomial sets via quasi monomiality the lowering operator s and the raising operator t associated with a polynomial set p n n 0 are two operators which are independent of n and satisfy the relationships s p n x np n 1 x and t p n x p n 1 x n n 0 in this paper we use these operators to study the orthogonality of some polynomial sets more precisely we state sufficient conditions in terms of the s and t operators to ensure the orthogonality we also express explicitly by means of the s operator the linear functional for which the orthogonality holds true we obtain some well known results as particular cases of the results presented in this paper

zeros of gegenbauer and hermite polynomials and connection coefficients in this paper sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of obrechkoff and certain explicit connection coefficients for these polynomials as a consequence sharp bounds for the zeros of the hermite polynomials are obtained

spatially periodic orbits in coupled map lattices we obtain the conditions that ensure the stability of spatially and temporally periodic orbits of coupled map lattices the stability matrices can be put in a circulant and block circulant form this allows us to reduce the problem to smaller matrices corresponding to the building blocks of spatial periodicity we find that additional conditions are imposed as we expand the size of the lattice for the traveling wave solution the analysis is considerably simplified we have analyzed both the one dimensional and higher dimensional lattices

efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of al salam carlitz i polynomials two formulae expressing explicitly the derivatives and moments of al salam carlitz i polynomials of any degree and for any order in terms of al salam carlitz i themselves are proved two other formulae for the expansion coefficients of general order derivatives d p f x and for the moments xd p f x of an arbitrary function f x in terms of its original expansion coefficients are also obtained application of these formulae for solving q difference equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between al salam carlitz i polynomials and any system of basic hypergeometric orthogonal polynomials belonging to the q hahn class is described

some remarks on a paper by l carlitz we study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by l carlitz we show that they are a special case of the sheffer polynomials and point out some interesting connections with certain sobolev orthogonal polynomials

solving connection and linearization problems within the askey scheme and its q analogue via inversion formulas for the polynomial families pn x n belonging to the askey scheme or to its q analogue the hypergeometric represen tation provides a natural expansion of the form pn x n 0 dm n m x where the expanding basism x is in general a product of pochhammer symbols or q shifted factorials in this paper we solve the corresponding inversion problem i e we compute the coe6cients im n in the expansionn x n m 0 im n pm x which are then used as a tool for solving any connection and linearization problem within the askey scheme and its q analogue extensions of this approach for polynomials outside these two schemes are also given c

quick construction of recurrence relations for the jacobi coefficients abstract recurrence relations are given for the jacobi series coefficients of functions which satisfy linear differential equations of first to third order with polynomial coefficients

some discrete d orthogonal polynomial sets in this paper we characterize all polynomial sets which are at the same time d orthogonal and appell the resulting polynomials reduce to charlier polynomials for d 1 1 various properties of the obtained polynomials are singled out generating function recurrence relation of order d 1 and a difference equation of order d 1 we also explicitly express the d dimensional functional for which the d orthogonality holds