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

liapunov schmidt reduction and continuation for nonlinear schro dinger equations we study the bifurcation scenario of nonlinear schro dinger equations nls the liapunov schmidt reduction is applied to show that the simple bifurcations of a single nls are pitchfork the pitchfork bifurcation can be subcritical or supercritical depending on the coefficient of the cubic term we choose we also describe numerical methods so that the liapunov schmidt reduction can effectively handle a corank 2 bifurcation point next we apply numerical continuation methods to trace solution curves and surfaces of the nls where the system is discretized by the centered difference approximations numerical results on two and three dimensional m coupled nls are reported where the physical properties such as the effect of trapping potentials isotropic and nonisotropic trapping potentials mass conservation constraints and strong and weak repulsive interactions are considered in our numerical experiments

gauss seidel type methods for energy states of a multi component bose einstein condensate in this paper we propose two iterative methods a jacobi type iteration ji and a gauss seidel type iteration gsi for the computation of energy states of the time independent vector gross pitaevskii equation vgpe which describes a multi component bose einstein condensate bec a discretization of the vgpe leads to a nonlinear algebraic eigenvalue problem naep we prove that the gsi method converges locally and linearly to a solution of the naep if and only if the associated minimized energy functional problem has a strictly local minimum the gsi method can thus be used to compute ground states and positive bound states as well as the corresponding energies of a multi component bec numerical experience shows that the gsi converges much faster than ji and converges globally within 10 20 steps

a cohen type inequality for laguerre sobolev expansions abstract let introduce the sobolev type inner product f g 0 f x g x d x m f 0 g 0 n f 0 g 0 where d x 1 1 x e x d x m n 0 1 in this paper we prove a cohen type inequality for the fourier expansion in terms of the orthonormal polynomials associated with the above sobolev inner product in particular for m n 0 we extend the result of markett c markett cohen type inequalities for jacobi laguerre and hermite expansions siam j math anal 14 1983 819 833

computing orthogonal polynomials in sobolev spaces numerical methods are considered for generating polynomials orthogonal with respect to an inner product of sobolev type i e one that involves derivatives up to some given order each having its own positive measure associated with it the principal objective is to compute the coefficients in the increasing order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next as a by product of this computation one gains access to the zeros of these polynomials via eigenvalues of an upper hessenberg matrix formed by the coefficients generated two methods are developed one is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as modified chebyshev algorithm the other a generalization of stieltjes s procedure expresses the desired coefficients in terms of a sobolev inner product involving the orthogonal polynomials in question whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point the numerical characteristics of these methods are illustrated in the case of sobolev orthogonal polynomials of old as well as new types based on extensive numerical experimentation a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros

varying discrete laguerre sobolev orthogonal polynomials asymptotic behavior and zeros we consider a varying discrete sobolev inner product involving the laguerre weight our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and of their zeros we are interested in mehler heine type formulas because they describe the asymptotic differences between these sobolev orthogonal polynomials and the classical laguerre polynomials moreover they give us an approximation of the zeros of the sobolev polynomials in terms of the zeros of other special functions we generalize some results appeared very recently in the literature for both the varying and non varying cases

generation and evaluation of orthogonal polynomials in discrete sobolev spaces i algorithms in this paper we study theoretically the determination and evaluation of polynomials that are orthogonal with respect to a general discrete sobolev inner product that is an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives this sobolev inner product has the property that the orthogonal polynomials with respect to it satisfy a linear recurrence relation of fixed order we provide a complete set of formulas to compute the coefficients of this recurrence besides we study the determination of the fourier sobolev coefficients of a finite approximation of a function and the numerical evaluation of the resulting finite series at a general point

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

zeros of orthogonal polynomials in certain discrete sobolev spaces abstract in this work discrete sobolev pseudo inner products of type 1 p q p c q c a b p x q x d x where d is a quasi definite borel measure and 0 and 2 p q p c q c p c q c a a p x q x d x where d is a symmetric positive borel measure c 0 and 0 are considered general properties of orthogonal polynomials associated with the above discrete sobolev inner products and their zeros are studied

differential equations of infinite order for sobolev type orthogonal polynomials abstract assume that pn x n 0 are orthogonal polynomials relative to a quasi definite moment functional which satisfy a differential equation of spectral type of order d 2 d where l i x are polynomials of degree i let be the symmetric bilinear form of discrete sobolev type defined by p q pq np k c q k c where n 0 and c are real constants k is a non negative integer and p and q are polynomials we first give a necessary and sufficient condition for to be quasi definite and then show if is quasi definite then the corresponding sobolev type orthogonal polynomials rnn k c x n 0 satisfy a differential equation of infinite order of the form p q np k c q k c where ai x i 0 are polynomials of degree i independent of n except a0 x a0 x n we also discuss conditions 3nder which such a differential equation is of finite order when is positive definite d

asymptotics for laguerre sobolev type orthogonal polynomials modified within their oscillatory regime abstract in this paper we consider sequences of polynomials orthogonal with respect to the discrete sobolev inner product f g s 0 f x g x x e x dx f c a g c t 1 where f and g are polynomials with real coefficients a r 2 2 and the vectors f c g c are a m 0 0 n f c f c f c and g c g c g c respectively with m n r and the mass point c is located inside the oscillatory region for the classical laguerre polynomials we focus our attention on the representation of these polynomials in terms of classical laguerre polynomials and we analyze the behavior of the coefficients of the corresponding five term recurrence relation when the degree of the polynomials is large enough also the outer relative asymptotics of the laguerre sobolev type with respect to the laguerre polynomials is analyzed

asymptotics of sobolev orthogonal polynomials for symmetrically coherent pairs of measures with compact support we study the strong asymptotics for the sequence of manic polynomials q c orthogonal with respect to the inner product u 3 9 s s f xmx h x 1 s f x s x 44x a 0 with x outside of the support of the measure 2 we assume that 1 and 2 are symmetric and compactly supported measures on lr satisfying a coherence condition as a consequence the asymptotic behaviour of q q s and of the zeros of qn is obtained

a nonterminating 8 φ 7 summation for the root system c r using multiple q integrals and a determinant evaluation we establish a nonterminating 8 7 summation for the root system cr we also give some important specializations explicitly

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

double centralizing theorems for the alternating groups let v n be the n fold tensor product of a vector space v following i schur we consider the action of the symmetric group sn on v n by permuting coordinates in the super z2 graded case v v0 v1 a sign is added these actions give rise to the corresponding schur algebras s sn v here s sn v is compared with s an v the schur algebra corresponding to the alternating subgroup an sn while in the classical signless case these two schur algebras are the same for n large enough it is proved that in the super case where dim v0 dim v1 s an v is isomorphic to the cross product algebra s an v s sn v z2

asymptotic behaviour of laguerre sobolev type orthogonal polynomials a nondiagonal case in this paper we study the asymptotic behaviour of polynomials orthogonal with respect to a sobolev type inner product s 0 p x q x x ae xdx p 0 taq 0 a 1 where p and q are polynomials with real coefficients a m 0 l lm 1 p 0 p 0 p 0 q 0 q 0 q 0 and a is a positive semidefinite matrix we will focus our attention on their outer relative asymptotics with respect to the standard laguerre polynomials as well as on an analog of the mehler heine formula for the rescaled polynomials

asymptotic behavior of varying discrete jacobi sobolev orthogonal polynomials in this contribution we deal with a varying discrete sobolev inner product involving the jacobi weight our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros we are interested in mehler heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these sobolev orthogonal polynomials and the jacobi ones moreover this asymptotic behavior provides an approximation of the zeros of the sobolev polynomials in terms of the zeros of other well known special functions we generalize some results appeared in the literature very recently

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

unbounded nilpotents and idempotents abstract this paper is concerned with unbounded operators in hilbert space with domain containing their range and in particular investigates unbounded nilpotents and idempotents

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

computing ground states of spin 1 bose einstein condensates by the normalized gradient flow in this paper we propose an efficient and accurate numerical method for computing the ground state of spin 1 bose einstein condensates becs by using the normalized gradient flow or imaginary time method the key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization this allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of a single component bec to compute the ground state of spin 1 becs an efficient and accurate discretization scheme the backward forward euler sine pseudospectral method is proposed to discretize the normalized gradient flow extensive numerical results on ground states of spin 1 becs with ferromagnetic antiferromagnetic interaction and harmonic optical lattice potential in one three dimensions are reported to demonstrate the efficiency of our new numerical method

on real and complex zeros of orthogonal polynomials in a discrete sobolev space abstract let sn x c n denote a set of polynomials orthogonal with respect to the discrete sobolev inner product f g f x g x d x nf c g c where n 0 c r for n 0 put kn x sn x 0 then sn x c n has at least n 2 different real zeros their position with respect to the zeros of kn can be determined using the tangent to the graph of y kn x in c kn c on the other hand if n 3 then c can be chosen such that sn x c n has two complex zeros if n is sufficiently large

a projection gradient method for computing ground state of spin 2 bose einstein condensates in this paper a projection gradient method is presented for computing ground state of spin 2 bose einstein condensates bec we first propose the general projection gradient method for solving energy functional minimization problem under multiple constraints in which the energy functional takes real functions as independent variables we next extend the method to solve a similar problem where the energy functional now takes complex functions as independent variables we finally employ the method into finding the ground state of spin 2 bec the key of our method is by constructing continuous gradient flows cgfs the ground state of spin 2 bec can be computed as the steady state solution of such cgfs we discretized the cgfs by a conservative finite difference method along with a proper way to deal with the nonlinear terms we show that the numerical discretization is normalization and magnetization conservative and energy diminishing numerical results of the ground state and their energy of spin 2 bec are reported to demonstrate the effectiveness of the numerical method

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

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

asymptotic behavior of the partial derivatives of laguerre kernels and some applications abstract the aim of this paper is to present some new results about the asymptotic behavior of the partial derivatives of the kernel polynomials associated with the gamma distribution we also show how these results can be used in order to obtain the inner relative asymptotics for certain laguerre sobolev type polynomials

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

zero distribution of orthogonal polynomials in a certain discrete sobolev space abstract we study the zero distribution of the polynomials snn which are orthogonal with respect to the discrete sobolev inner product f g 0 x g x d x nf r 0 g r 0 where is a distribution function n 0 r 1 snn has n real simple zeros at most one of them is outside 0 the location of these zeros is given in relation to the position of the zeros of some classical polynomials i e polynomials with respect to an inner product with n 0

efficient numerical methods for computing ground states of spin 1 bose einstein condensates based on their characterizations in this paper we propose efficient numerical methods for computing ground states of spin 1 bose einstein condensates becs with without the ioffe pritchard magnetic field b x when b x 0 a numerical method is introduced to compute the ground states and it is also applied to study properties of ground states numerical results suggest that the densities of m f 1 components in ground states are identical for any nonzero b x in particular if b x b 0 is a constant the ground states satisfy the single mode approximation when b x 0 efficient and simpler numerical methods are presented to solve the ground states of spin 1 becs based on their ferromagnetic antiferromagnetic characterizations numerical simulations show that our methods are more efficient than those in the literature in addition some conjectures are made from our numerical observations

discrete laguerre sobolev expansions a cohen type inequality abstract c markett proved a cohen type inequality for the classical laguerre expansions in the appropriate weighted l p spaces in this paper we get a cohen type inequality for the fourier expansions in terms of discrete laguerre sobolev orthonormal polynomials with an arbitrary finite number of mass points so we extend the result due to b xh fejzullahu and f marcellan

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

global properties of zeros for sobolev type orthogonal polynomials perez t e and m a pifiar global properties of zeros for sobolev type orthogonal polynomials journal of computational and applied mathematics 49 1993 22 5 232 in this paper we analyze some properties concerning the zeros of orthogonal polynomials q x associated to the inner product f g f xlg x dp x1 t mf c g c nf c g c where i is a not necessarily bounded real interval p is a positive measure on i c e w and m n 2 0 in particular some properties concerning the localization and separation for the roots of q x are obtained

some inequalities for the q beta and the q gamma functions via some q integral inequalities some new inequalities for the q gamma the q beta and the q analogue of the psi functions are established via some q integral inequalities

a tool for locating zeros of orthogonal polynomials in sobolev inner product spaces abstract in the theory of polynomials orthogonal with respect to an inner product of the form f 0f x g x d x mk 1akf ik 0 g ik 0 one is confronted with the following situation for certain values of the parameters the orthogonal polynomial of degree n does not have all its zeros inside the support of the distribution function d this paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial for the case m 2 it is shown that there are exactly two zeros outside the true interval of orthogonality for a1 a2 large moreover it is proved that these zeros are nonreal complex conjugates in the case i1 1 i2 also several examples are given

asymptotics for varying discrete sobolev orthogonal polynomials abstract we consider a varying discrete sobolev inner product such as f g s f x g x d m n f j c g j c where is a finite positive borel measure supported on an infinite subset of the real line c is adequately located on the real axis j 0 and mn n 0 is a sequence of nonnegative real numbers satisfying a very general condition our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this sobolev inner product in this way we focus our attention on mehler heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c just the point where we have located the perturbation of the standard inner product moreover we pay attention to the asymptotic behavior of the scaled zeros of these varying sobolev polynomials and some numerical experiments are shown finally we provide other asymptotic results which strengthen the idea that mehler heine asymptotics describe in a precise way the differences between sobolev orthogonal polynomials and standard ones

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

connection formulas for general discrete sobolev polynomials in this paper the discrete sobolev inner product p q p x q x d i 0 r m i p i c q i c is considered where is a finite positive borel measure supported on an infinite subset of the real line c r and mi 0 i 0 1 r connection formulas for the orthonormal polynomials associated with are obtained as a consequence for a wide class of measures we give the mehler heine asymptotics in the case of the point c is a hard edge of the support of in particular the case of a symmetric measure is analyzed finally some examples are presented

a spectral galerkin continuation method for numerical solutions of the gross pitaevskii equation abstract we study spectral galerkin methods sgm for nonlinear eigenvalue problems where the legendre polynomials are used as the basis functions for the trial function space the sgm is applied to find the ground state solution of the gross pitaevskii equation gpe and the gpe in a periodic potential when the sgm is incorporated in the context of continuation methods for curve tracking a manifest advantage is that the target points are very close to the bifurcation points thus it takes less continuation steps to reach the target point when compared with the centered difference methods cdm or finite element methods fem we also implement the spectral collocation method scm so that the computational cost for the numerical integration in the sgm can be saved comprehensive numerical experiments on the gpe using various numerical methods described in this paper are reported

composition operators on hilbert spaces of entire functions with analytic symbols abstract composition operators with analytic symbols on some reproducing kernel hilbert spaces of entire functions on a complex hilbert space are studied the questions of their boundedness seminormality and positivity are investigated it is proved that if such an operator is bounded then its symbol is a polynomial of degree at most 1 i e it is an affine mapping fock s type model for composition operators with linear symbols is established as a consequence explicit formulas for their polar decomposition aluthge transform and powers with positive real exponents are provided the theorem of carswell maccluer and schuster is generalized to the case of segal bargmann spaces of infinite order some related questions are also discussed

on suboperators with codimension one domains the question of the existence of a proper closed densely defined suboperator whose domain contains a given linear space is answered such suboperators with codimension one domains are described the motivation for studying this topic comes from the theory of extremal extensions of non densely defined positive operators

efficient and spectrally accurate numerical methods for computing ground and first excited states in bose einstein condensates in this paper we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in bose einstein condensates becs we begin with a review on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the gfdn for discretizing the gradient flow we use sine pseudospectral method for spatial derivatives and either backward euler scheme besp or backward forward euler schemes for linear nonlinear terms bfsp for temporal derivatives both besp and bfsp are spectral order accurate for computing the ground and first excited states in bec of course they have their own advantages i for linear case besp is energy diminishing for any time step size where bfsp is energy diminishing under a constraint on the time step size ii at every time step the linear system in bfsp can be solved directly via fast sine transform fst and thus it is extremely efficient and in besp it needs to be solved iteratively via fst by introducing a stabilization term and thus it could be efficient too comparisons between besp and bfsp as well as other existing numerical methods are reported in terms of accuracy and total computational time our numerical results show that both besp and bfsp are much more accurate and efficient than those existing numerical methods in the literature finally our new numerical methods are applied to compute the ground and first excited states in bec in one dimension 1d 2d and 3d with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution