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

lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

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

generalizations of laguerre polynomials it is shown that the polynomials ln m0 m1 mn x n 0 defined by ln m0m1 mn x k 0n 1ak dkln x for certain real coefficients ak k 0n 1 are orthogonal with respect to the inner product f g 1 1 0 x e x f x g x dx v 0nmv f v 0 g v 0 where 1 n n and mv 0 for all v 0 1 2 n for these new polynomials n m0 m1 mn x n 0 an orthogonality relation and a second order differential equation are derived further we obtain a representation as a n 2fn 2 hypergeometric series and a 2n 3 terms recurrence relation which gives rise to a christoffel darboux type formula

expansions in series of varying laguerre polynomials and some applications to molecular potentials the expansion of a large class of functions in series of linearly varying laguerre polynomials i e laguerre polynomials whose parameters are linear functions of the degree is found by means of the hypergeometric functions approach this expansion formula is then used to obtain the brown carlitz generating function which gives a characterization of the exponential function and the connection formula for these polynomials finally these results are employed to connect the bound states of the quantum mechanical potentials of morse and poschl teller which are frequently used to describe molecular systems

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

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

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable on the lattice x s q 2 abstract we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable on a non uniform lattice x s q2s i e the q analogues of charlier meixner krawtchouk and hahn polynomials in terms of the coefficients and of the pearson equation satisfied by the weight function and the coefficients of the three term recurrence relation and of two structure relations obeyed by these polynomials

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

a basic class of symmetric orthogonal polynomials using the extended sturm liouville theorem for symmetric functions abstract in this research by applying the extended sturm liouville theorem for symmetric functions a basic class of symmetric orthogonal polynomials bcsop with four free parameters is introduced and all its standard properties such as a generic second order differential equation along with its explicit polynomial solution a generic orthogonality relation a generic three term recurrence relation and so on are presented then it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class they are respectively the generalized ultraspherical polynomials generalized hermite polynomials and two other sequences of symmetric polynomials which are finitely orthogonal on and can be expressed in terms of the mentioned class directly in this way two half trigonometric sequences of orthogonal polynomials as special sub cases of bcsop are also introduced

properties of zeros of orthogonal polynomials and related functions abstract there are many results in the literature on orthogonal polynomials concerning the way in which the zeros of a polynomial change as a parameter changes for the classical polynomials there are results due to markoff 1886 and stieltjes 1886 on the monotonic variation of zeros of jacobi and gegenbauer ultraspherical polynomials with respect to parameters our purpose here is to describe some recent results concerning formulas for the derivatives of zeros with respect to a parameter such formulas and their consequences are fairly well developed for bessel functions a limiting case of some of the classical orthogonal polynomials and have recently been explored for some orthogonal polynomials and related functions two general methods will be described the first use extensively by ismail the author and others is based on the hellmann feynman theorem which gives the derivative of an eigenvalue with respect to a parameter it can be applied in either a differential equations or a recurrence relations setting the second method developed in recent work of elbert and the author and applicable to suitable solutions of second order ordinary linear differential equations originates in work of richardson 1918 from this approach we obtain formulas for dc d where c c is a zero of the ultraspherical polynomial p n x these recover the known result that dc d 1 2 we get similar results for certain hermite functions these results are closely related to nicholson type formulas due to durand 1975 for sums of squares of solutions of second order linear ordinary differential equations nicholson type formulas are named after the corresponding formulas for bessel functions due to nicholson

on affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures we consider two positive normalized measures da x and db x related by the relationship da x cx ddb x or by da x cx 2 edb x and db x is symmetric we show that then the polynomial sequences a n x b n x orthogonal with respect to these measures are related by the relationship a n x b n x email protected nb n 1 x or by a n x b n x email protected nb n 2 x for some sequences k n and l n we present several examples illustrating this fact and also present some attempts for extensions and generalizations we also give some universal identities involving polynomials b n x and the sequence k n that have a form of the fourier series expansion of the radon nikodym derivative of one measure with respect to the other

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 and zeros of orthogonal polynomials abstract we discuss an old theorem of obrechkoff and some of its applications some curious historical facts around this theorem are presented we make an attempt to look at some known results on connection coefficients zeros and wronskians of orthogonal polynomials from the perspective of obrechkoff s theorem necessary conditions for the positivity of the connection coefficients of two families of orthogonal polynomials are provided inequalities between the kth zero of an orthogonal polynomial pn x and the largest smallest zero of another orthogonal polynomial qn x are given in terms of the signs of the connection coefficients of the families pn x and qn x an inequality between the largest zeros of the jacobi polynomials pn a b x and pn x is also established

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

a set of finite order differential equations for the appell polynomials let r n x n 0 denote the set of appell polynomials which includes among others hermite bernoulli euler and genocchi polynomials in this paper by introducing the generalized factorization method for each k n we determine the differential operator l n k x n 0 such that l n k x r n x l n kr n x where l n k n k n k the special case k 1 reduces to the result obtained in m x he p e ricci differential equation of appell polynomials via the factorization method j comput appl math 139 2002 231 237 the differential equations for the hermite and bernoulli polynomials are exhibited for the case k 2

explicit formula for the multiple generating function of product of generalized laguerre polynomials using an operator approach messina and paladimo have recently demonstrated a method of evaluating a multiple generating function of the product of generalized laguerre polynomials and arrived at the solution which is given implicitly by using an addition theorem of vilenkin for the laguerre polynomials which has its origin in group theory we are able to give an explicit formula for the multiple generating function in terms of elementary symmetric functions it is further shown that the multiple generating function is a symmetric function an interesting property which is not altogether obvious from the solution via the operator approach

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

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

positive semidefinite rank let m in mathbb r p times q m rp q be a nonnegative matrix the positive semidefinite rank psd rank of m is the smallest integer k for which there exist positive semidefinite matrices a i b j ai bj of size k times k k k such that m ij mathrm trace a i b j mij trace aibj the psd rank has many appealing geometric interpretations including semidefinite representations of polyhedra and information theoretic applications in this paper we develop and survey the main mathematical properties of psd rank including its geometry relationships with other rank notions and computational and algorithmic aspects

linearization coefficients for boas buck polynomial sets abstract in this paper we express explicitly the linearization coefficients related to three boas buck polynomial sets using their corresponding generating functions we apply the obtained results to many classes of polynomials including some generalized hypergeometric polynomials the corresponding linearization coefficients will be expressed by means of terminating generalized kampe de feriet functions which for some cases can be reduced to hypergeometric functions or hypergeometric terms using with this purpose some known reduction formulae we obtain some well known results as particular cases as well as some new reduction formulae

a discrete approach to monotonicity of zeros of orthogonal polynomials we study the monotonicity with respect to a parameter of zeros of orthogonal polynomials our method uses the tridiagonal jacobi matrices arising from the three term recurrence relation for the polynomials we obtain new results on monotonicity of zeros of associated laguerre al salam carlitz meixner and pollaczek polynomials we also derive inequalities for the zeros of the al salam carlitz and meixner polynomials

hopf bifurcation at 1 1 resonance on etudie un probleme concernant la bifurcation de hopf pour une famille a 1 parametre d equations differentielles dans un cas de resonance

strengthening convex relaxations of 0 1 sets using boolean formulas in convex integer programming various procedures have been developed to strengthen convex relaxations of sets of integer points on the one hand there exist several general purpose methods that strengthen relaxations without specific knowledge of the set s of feasible integer points such as popular linear programming or semi definite programming hierarchies on the other hand various methods have been designed for obtaining strengthened relaxations for very specific sets s that arise in combinatorial optimization we propose a new efficient method that interpolates between these two approaches our procedure strengthens any convex set containing a set s subseteq 0 1 n by exploiting certain additional information about s namely the required extra information will be in the form of a boolean formula phi defining the target set s the new relaxation is obtained by feeding the convex set into the formula phi we analyze various aspects regarding the strength of our procedure as one application interpreting an iterated application of our procedure as a hierarchy our findings simplify improve and extend previous results by bienstock and zuckerberg on covering problems

laguerre type special functions and population dynamics abstract we introduce new laguerre type population dynamics models these models arise quite naturally by substituting in classical models the ordinary derivatives with the laguerre derivatives and therefore by using the so called laguerre type exponentials instead of the ordinary exponential the l exponentials e n t are increasing convex functions for t 0 but increasing slower with respect to exp t for this reason these functions are useful in order to approximate different behaviors of population growth we consider mainly the laguerre type derivative d t t d t connected with the l exponential e 1 t and investigate the corresponding modified logistic bernoulli and gompertz models invariance of the volterra lotka model is mentioned

second order recurrence relation for the linearization coefficients of the classical orthogonal polynomials abstract let pk be any sequence of the classical orthogonal polynomials we give explicitly a second order recurrence relation in k for the coefficients in p i p j k i j i j c i j k p k this result is obtained by a method which is alternative to the one proposed recently by a ronveaux et al applications of the result to some named systems of classical orthogonal polynomials are given

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

fourth order resonant collisions of multipliers in reversible maps period 4 orbits and invariant curves resonant collision of multipliers at i of a symmetric fixed point for a 2 parameter family of 4 dimensional reversible maps is considered bifurcation of period 4 orbits from the fixed point and their linear stability characteristics are briefly reviewed in one of the three possible types of bifurcation see text a small angle secondary collision of the floquet multipliers of the bifurcating periodic orbit takes place leading to the bifurcation of invariant curves from the orbit the invariant curves are calculated in a perturbation scheme in the leading order of perturbation the secondary bifurcation is found to be of superthreshold type an interesting pattern in the vicinity of the resonant collision involving families of invariant curves and 2 tori emerges results of numerical iterations corroborating the picture conjectured on the basis of perturbation calculations are presented corresponding results on resonant collisions at 1 are briefly stated

d orthogonality of hermite type polynomials abstract in this paper we state a characterization theorem for d orthogonal polynomials of hermite type as application we solve two characterization problems related to gould hopper polynomials and charlier type polynomials various properties of the resulting generalized hypergeometric polynomials are singled out in particularly we derive the d dimensional functional vectors ensuring the d orthogonality of these polynomials the obtained weight functions are reduced to the well known weight functions of charlier and hermite polynomials for d 1 we also explicitly express the d dimensional functional vector associated with polynomials of laguerre type which are related to the obtained hermite type polynomials

some 0 1 polytopes need exponential size extended formulations we prove that there are 0 1 polytopes p subseteq mathbb r n that do not admit a compact lp formulation more precisely we show that for every n there is a set x subseteq 0 1 n such that conv x must have extension complexity at least 2 n 2 cdot 1 o 1 in other words every polyhedron q that can be linearly projected on conv x must have exponentially many facets in fact the same result also applies if conv x is restricted to be a matroid polytope conditioning on mathbf np not subseteq mathbf p poly our result rules out the existence of a compact formulation for any mathbf np hard optimization problem even if the formulation may contain arbitrary real numbers

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

an operator approach to the construction of generating functions for products of associated laguerre polynomials we introduce an iteratively defined operator sequence allowing the construction of new useful mathematical identities involving the product of associated laguerre polynomials possible physical applications as well as some methodological aspects of our approach are pointed out

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

smallest compact formulation for the permutahedron in this note we consider the permutahedron the convex hull of all permutations of 1 2 ldots n 1 2 n we show how to obtain an extended formulation for this polytope from any sorting network by using the optimal ajtai komlos szemeredi sorting network this extended formulation has vartheta n log n nlogn variables and inequalities furthermore from basic polyhedral arguments we show that this is best possible up to a multiplicative constant since any extended formulation has at least varomega n log n nlogn inequalities the results easily extend to the generalized permutahedron

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

convex hulls of curves of genus one abstract let c be a real nonsingular affine curve of genus one embedded in affine n space whose set of real points is compact for any polynomial f which is nonnegative on c r we prove that there exist polynomials f i with f i f i 2 mod i c and such that the degrees deg f i are bounded in terms of deg f only using lasserre s relaxation method we deduce an explicit representation of the convex hull of c r in r n by a lifted linear matrix inequality this is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety the same works for convex hulls of singular curves whose normalization is c we then make a detailed study of the associated degree bounds these bounds are directly related to size and dimension of the projected matrix pencils in particular we prove that these bounds tend to infinity when the curve c degenerates suitably into a singular curve and we provide explicit lower bounds as well

laguerre type exponentials and generalized appell polynomials general classes of two variables appell polynomials are introduced by exploiting properties of an iterated isomorphism related to the so called laguerre type exponentials further extensions to the multi index and multivariable cases are mentioned

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

generating functions via integral transforms abstract in this paper we use some integral transforms to derive for a polynomial sequence p n x n 0 generating functions of the type g x t n 0 n p n x t n starting from a generating function of type g x t n 0 p n x t n where n n 0 is a real numbers sequence independent on x and t that allows us to unify the treatment of a generating function problem for many well known polynomial sequences in the literature

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

d orthogonality of humbert and jacobi type polynomials in this paper we treat three questions related to the d orthogonality of the humbert polynomials the first one consists to determinate the explicit expression of the d dimensional functional vector for which the d orthogonality holds the second one is the investigation of the components of humbert polynomial sequence that allows us to introduce as far as we know new d orthogonal polynomials generalizing the classical jacobi ones the third one consists to solve a characterization problem related to a generalized hypergeometric representation of the humbert polynomials

stability of invariant curves in four dimensional reversible mappings near 1 1 resonance recent work on reversible neimark sacker bifurcation in 4d maps is summarized linear stability analysis of families of invariant curves appearing in this bifurcation is presented by a referring to the analogous stability problem in reversible hopt bifurcation in vector fields and b perturbatively calculating a set of quantities termed quasi multipliers for the invariant curves in particular the critical rotation numbers corresponding to transition from elliptic to hyperbolic invariant curves on the subthreshold side in the so called inverted bifurcation are calculated results of numerical iterations corroborating the above analysis are presented the question of exploring the structure of the phase space close to the invariant curves is briefly addressed in the conclusion

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

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

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

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

higher order hypergeometric lauricella function and zero asymptotics of orthogonal polynomials the asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a lauricella function the polynomials are defined by means of a three term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices subclasses of systems of orthogonal polynomials having their contracted measure of zeros of regular uniform wigner weyl karamata and hypergeometric types are explicitly identified some illustrative examples are given

a characterization of dunkl classical d symmetric d orthogonal polynomials and its applications abstract in this paper we obtain a necessary and sufficient condition on a linear operator j defined on polynomials and a d symmetric d orthogonal polynomial set p n n 0 such that j p n n 0 is also d orthogonal that allows us to characterize the dunkl classical d symmetric d orthogonal polynomials as the range of classical d symmetric d orthogonal polynomials by an operator related to the dunkl operator as applications we derive many properties of the dunkl classical d symmetric d orthogonal polynomials from the classical d symmetric d orthogonal polynomials

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

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

on the existence of 0 1 polytopes with high semidefinite extension complexity in rothvo math program 142 1 2 255 268 2013 it was shown that there exists a 0 1 polytope a polytope whose vertices are in 0 1 n 0 1 n such that any higher dimensional polytope projecting to it must have 2 varomega n 2 n facets i e its linear extension complexity is exponential the question whether there exists a 0 1 polytope with high positive semidefinite extension complexity was left open we answer this question in the affirmative by showing that there is a 0 1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2 varomega n 2 n and an affine space our proof relies on a new technique to rescale semidefinite factorizations

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

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

recurrence relations for hypergeometric functions of unit argument we show that the generalized hypergeometric function p 1 p 3fp 2 nn 2 1 a n 0 satisfies a nonhomogeneous recurrence relation of order p a where a 0 when p 3 2 1 is balanced and a 1 otherwise also for us dq n a 4 2 q l n dq 2n a 1 1 n o a homogeneous recurrence relation of order q 1 is given 1 notation the generalized hypergeometric series a a l a 1 00 p 1 p 1 1 p t ail b p e al ai kz k k h b1 k where 1 2 a k f a k f a converges for iz i 0 let x ai i 1 2 p b1 j 1 2 p 2 ck k 1 2 q 2 di i 1 2 q be complex constants we define the functions g 1 n t n 1 m m 2n x 2m 2m n a m m 2n x m t m n bp 21 x m n r 1 bp 2 3 1 f m 2n x m t n m bp 2 p 4p 3l 2n x 2m 1 n m 1 bp 2 f 2n a m t n m ap 1 m 2n x t 1 n m 1 a 2n x 2m 1 n m a j h 3 n t 2n x m n b 1 m n x m s 1 m n 2n a n 1 cq t m q 4 q 3 2n a q 2 n cq 2 3 2 2n x m n dq rn m n cq 2 1 m l2n m 1 n 1 dq xq 2 q l 2n x q 2 n dq here b 1 ci for i q 2 we assume the parameters to be such that all expressions make sense by applying a lemma of wimp 8 we get the alternative forms for gm hm gm n t l p n 1 m n 2n x 2m 2n x t i t n x m m 2n x m t m n 1 bp 2 x t r n x t 1 2 r t 2n x m t n x t 2 bp 2 p 4p3 2n x t 1 n x t 1 bp 2 2n x m t n x t 1 ap f t rn mr t 1 2n x m t 1 n x t 2 a xp 2fp l2 2n x t 1 n x t 1 ap j this content downloaded from 157 55 39 116 on sun 18 sep 2016 06 20 49 utc all use subject to http about jstor org terms 524 stanislaw lewanowicz l1 q 2n x q 2 n 1 m n x m 2n x m mi n cq 2

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

on the extension complexity of combinatorial polytopes in this paper we extend recent results of fiorini et al on the extension complexity of the cut polytope and related polyhedra we first describe a lifting argument to show exponential extension complexity for a number of np complete problems including subset sum and three dimensional matching we then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs including those used in quantum information and suspensions of cubic planar graphs

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

differential equations satisfied by the components with respect to the cyclic group of order n of some special functions let f be a complex function of the variable z admitting a laurent expansion in an annulus c with center in the origin for an arbitrary positive integer n ricci s theorem asserts that the function f can be written as the sum of n functions f n k k 0 1 n 1 defined byf n k z 1n l 0n 1exp 2i klnfzexp2i ln z c in this paper we present a technique which starting from a suitable differential equation satisfies by the function f provides a differential equation satisfied by the functions f n k some known special functions are treated for illustration

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

maximum semidefinite and linear extension complexity of families of polytopes we relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise hausdorff distance of its members this result directly implies a known lower bound on the maximum semidefinite extension complexity of 0 1 polytopes we further show how our result can be used to improve on the corresponding bounds known for polygons with integer vertices our geometric proof builds upon nothing else than a simple well known property of maximum volume inscribed ellipsoids of convex bodies in particular it does not rely on factorizations over the semidefinite cone and thus avoids involved procedures of balancing them as required e g in briet et al math program 153 1 179 199 2015 moreover we show that the linear extension complexity of every d dimensional 0 1 polytope is bounded from above by o frac 2 d d

linearization and connection coefficients for hypergeometric type polynomials we consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric type polynomials directly in terms of the corresponding differential equations we illustrate the method by producing explicit formulas for hermite polynomials

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

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

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

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

the relation of the d orthogonal polynomials to the appell polynomials abstract we are dealing with the concept of d dimensional orthogonal abbreviated d orthogonal polynomials that is to say polynomials verifying one standard recurrence relation of order d 1 among the d orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials in particular we are concerned in the present paper with the solution of the following problem p find all polynomial sequences which are at the same time appell polynomials and d orthogonal the resulting polynomials are a natural extension of the hermite polynomials a sequence of these polynomials is obtained all the elements of its d 1 order recurrence are explicitly determined a generating function a d 1 order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given among such polynomials one singles out the d symmetrical ones definition 1 7 which are the d orthogonal polynomials analogous to the hermite classical ones when d 1 ordinary orthogonality we meet again the classical orthogonal polynomials of hermite

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

classification and unfoldings of 1 2 resonant hopf bifurcation in this paper we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1 2 ratio this corresponds to a hopf hopf mode interaction with 1 2 resonance as occurs in the context of dissipative mechanical systems using an approach based on liapunov schmidt reduction and singularity theory we give a framework in which to study these problems and their perturbations in two cases no distinguished parameter and one distinguished bifurcation parameter we give a complete classification of the generic cases and their unfoldings

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

d orthogonality via generating functions in this paper starting from a suitable generating function of a polynomial set we show how to decide whether the considered polynomial set is d orthogonal and if it is so how to determine the corresponding d dimensional functional vector then we apply the obtained results to some known and new d orthogonal polynomial sets for the known ones we give new proofs for some already obtained results

difference equations of q appell polynomials in this paper we study some properties of the q appell polynomials including the recurrence relations and the q difference equations which extend some known classical q 1 results we also provide the recurrence relations and the q difference equations for q bernoulli polynomials q euler polynomials q genocchi polynomials and for q hermite polynomials as special cases of q appell polynomials

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