notes on a conservative nonlinear oscillator the amplitude frequency relationship is an important mathematical property for a nonlinear oscillator he s amplitude frequency formulation and the max min approach are used to handle the conservative nonlinear oscillator email protected email protected 2 x 0 for the amplitude frequency relationship the obtained result is compared with those in the open literature revealing the effectiveness of the used methods

sylvester equations for laguerre hahn orthogonal polynomials on the real line matrix sylvester differential equations are introduced in the study of laguerre hahn orthogonal polynomials matrix sylvester differential systems are shown to yield representations for the laguerre hahn orthogonal polynomials lax pairs are given formed from the differential system and the recurrence relation that yield discrete non linear equations for the three term recurrence relation coefficients of the laguerre hahn orthogonal polynomials

atomic and molecular decompositions of anisotropic besov spaces in this work we develop the theory of weighted anisotropic besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms this study extends the isotropic littlewood paley methods of dyadic transforms of frazier and jawerth 19 21 to non isotropic settings

factorization of the q difference equation for continuous q jacobi polynomials the formalism of factorization recently used by atakishiyev et al 2007 j phys a math theor 40 9311 7 to study the q difference equation for continuous q hermite polynomials is extended to the continuous q jacobi polynomials particular cases of continuous q laguerre q hermite q legendre and q ultraspherical polynomials are easily recovered

deformed laguerre hahn orthogonal polynomials on the real line abstract one derives discrete dynamical systems related to laguerre hahn orthogonal polynomials one studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t dependence on the coefficients of the riccati differential equation for the related stieltjes function

characterization theorem for laguerre hahn orthogonal polynomials on non uniform lattices abstract a characterization theorem for laguerre hahn orthogonal polynomials on non uniform lattices is stated and proved this theorem proves the equivalence between the riccati equation for the formal stieltjes function linear first order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind and linear first order difference relations for the functions of the second kind

generalizations of orthogonal polynomials we give a survey of recent generalizations of orthogonal polynomials that includes multidimensional matrix and vector orthogonal polynomials and multivariate versions multipole orthogonal rational functions variants and extensions of the orthogonality conditions multiple orthogonality most of these generalizations are inspired by the applications in which they are applied we also give a glimpse of these applications which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role moment problems numerical quadrature rational approximation linear algebra recurrence relations and random matrices

characterizations of weighted besov and triebel lizorkin spaces via temperatures abstract hardy littlewood type characterizations via temperatures on a half space of weighted besov and triebel lizorkin spaces studied recently by the author are given in the proof of the sufficient parts of the results a sub mean value property of temperatures is used in an elegant way to get control of certain terms by the hardy maximal function or its vector valued version as a byproduct of the results another characterization of weighted hardy spaces as well as characterizations of the gauss weierstrass integrals on a strip domain of distributions in the above weighted spaces is obtained also

pade approximation and apostol bernoulli and apostol euler polynomials using the pade approximation of the exponential function we obtain recurrence relations between apostol bernoulli and between apostol euler polynomials as applications we derive some new lacunary recurrence relations for bernoulli and euler polynomials with gap of length 4 and lacunary relations for bernoulli and euler numbers with gap of length 6

a characterization of hajlasz sobolev and triebel lizorkin spaces via grand littlewood paley functions abstract in this paper we establish the equivalence between the hajlasz sobolev spaces or classical triebel lizorkin spaces and a class of grand triebel lizorkin spaces on euclidean spaces and also on metric spaces that are both doubling and reverse doubling in particular when p n n 1 we give a new characterization of the hajlasz sobolev spaces m 1 p r n via a grand littlewood paley function

decompositions of besov morrey spaces and triebel lizorkin morrey spaces the aim of this paper is to define the besov morrey spaces and the triebel lizorkin morrey spaces and to present a decomposition of functions belonging to these spaces our results contain an answer to the conjecture proposed by mazzucato

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

pointwise characterizations of besov and triebel lizorkin spaces and quasiconformal mappings abstract in this paper the authors characterize in terms of pointwise inequalities the classical besov spaces b p q s and triebel lizorkin spaces f p q s for all s 0 1 and p q n n s both in r n and in the metric measure spaces enjoying the doubling and reverse doubling properties applying this characterization the authors prove that quasiconformal mappings preserve f n s q s on r n for all s 0 1 and q n n s a metric measure space version of the above morphism property is also established

structure relations for the bivariate big q jacobi polynomials we give structure relations for the orthogonal polynomials in two variables defined by lewanowicz and wozny s lewanowicz p wozny j comput appl math 233 2010 1554 1561 p n k x y a b c d q p n k y a bcq 2 k 1 dq k q y kdq y q kp kx y c b d y q n 0 k 0 1 n where email protected 0 1 0

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

homotopy perturbation method for a conservative x1 3 force nonlinear oscillator the homotopy perturbation method is used to obtain the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x 1 3 we find this method works very well for the whole range of initial amplitudes excellent agreement of the approximate frequencies with the exact ones has been demonstrated and discussed only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 0 60 for small and large values of oscillation amplitude while this relative error is as low as 0 050 for the second iteration comparison of the results obtained using this method with those obtained by different harmonic balance methods reveals that the former is more effective and convenient for these types of nonlinear oscillators

atomic and molecular decompositions of anisotropic triebel lizorkin spaces weighted anisotropic triebel lizorkin spaces are introduced and studied with the use of discrete wavelet transforms this study extends the isotropic methods of dyadic transforms of frazier and jawerth 1985 1989 to non isotropic settings associated with general expansive matrix dilations and a weights in close analogy with the isotropic theory we show that weighted anisotropic triebel lizorkin spaces are characterized by the magnitude of the transforms in appropriate sequence spaces we also introduce non isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces thus extending isotropic results of frazier and jawerth

hardy spaces with variable exponents and generalized campanato spaces in the present paper we define hardy spaces with variable exponents on rn by the grand maximal function and then investigate their several properties the present paper will connect harmonic analysis with function spaces with variable exponents we obtain the atomic decomposition and the molecular decomposition with these decomposition proved we investigate the littlewood paley characterization also we specify the dual spaces of hardy spaces with variable exponents they will turn out to be campanato spaces with variable growth conditions the present paper covers local hardy spaces with variable exponents

on orthogonal polynomials with perturbed recurrence relations abstract orthogonal polynomials may be fully characterized by the following recurrence relation pn x x n 1 pn 1 x n 1pn 2 x with p0 x 1 p1 x x 0 and n 0 here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the and parameters of a co recursive k k co dilated k k and co modified k k k k nature for an arbitrary but fixed kth element 1 k specifically stieltjes functions differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given this type of problems is strongly related to the boundary value problems of finite difference equations and to the quantum mechanical study of physical many body systems atoms molecules nuclei and solid state systems

linear partial q difference equations on q linear lattices and their bivariate q orthogonal polynomial solutions orthogonal polynomial solutions of an admissible potentially self adjoint linear second order partial q difference equation of the hypergeometric type in two variables on q linear lattices are analyzed a q pearson s system for the orthogonality weight function as well as for the difference derivatives of the solutions are presented giving rise to a solution of the q difference equation under study in terms of a rodrigues type formula the monic orthogonal polynomial solutions are treated in detail giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy lewanowicz and wozny s lewanowicz p wozny j comput appl math 233 2010 1554 1561 have recently introduced a non monic bivariate extension of big q jacobi polynomials together with a partial q difference equation of the hypergeometric type that governs them this equation is analyzed in the last section we provide two more orthogonal polynomial solutions namely a second non monic solution from the rodrigues representation and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series limit relations as email protected for the partial q difference equation and for the all three q orthogonal polynomial solutions are also presented

a direct proof of the christoffel darboux identity and its equivalence to the recurrence relationship abstract first a direct proof of the christoffel darboux identity for orthogonal polynomials is given this proof does not make use of the three terms recurrence relationship then it is proved that if a family of polynomials satisfies the christoffel darboux identity then it also satisfies a three terms recurrence relationship and thus it forms a family of orthogonal polynomials which shows the equivalence between both relations

decompositions of besov hausdorff and triebel lizorkin hausdorff spaces and their applications abstract let p 1 q 1 s r and 0 1 1 max p q in this paper the authors establish the transform characterizations of besov hausdorff spaces b h p q s r n and triebel lizorkin hausdorff spaces f h p q s r n q 1 as applications the authors then establish their embedding properties which on b h p q s r n is also sharp smooth atomic and molecular decomposition characterizations for suitable moreover using their atomic and molecular decomposition characterizations the authors investigate the trace properties and the boundedness of pseudo differential operators with homogeneous symbols in b h p q s r n and f h p q s r n q 1 which generalize the corresponding classical results on homogeneous besov and triebel lizorkin spaces when p 1 and q 1 by taking 0

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

a new class of function spaces connecting triebel lizorkin spaces and q spaces abstract let s r 0 p 1 and q 1 in this paper we introduce a new class of function spaces f p q s r n which unify and generalize the triebel lizorkin spaces with both p 1 and p and q spaces by establishing the carleson measure characterization of q space we then determine the relationship between triebel lizorkin spaces and q spaces which answers a question posed by dafni and xiao in g dafni j xiao some new tent spaces and duality theorem for fractional carleson measures and q r n j funct anal 208 2004 377 422 moreover via the hausdorff capacity we introduce a new class of tent spaces f t p q s r n 1 and determine their dual spaces f w p q s q r n where s r p q 1 max p q 1 0 q max p q and t denotes the conjugate index of t 1 as an application of this we further introduce certain hardy hausdorff spaces f h p q s r n and prove that the dual space of f h p q s r n is just f p q s q r n when p q 1

on a problem of modified lindstedt poincare for certain strongly non linear oscillators in the present paper concerns the modified lindstedt poincare method is presented for extending the rang of the validity of perturbation expansion to strongly non linear oscillation it is an effective method for dealing with strongly non linear vibration of single degree of freedom system which cannot be treated by the standard l p method therefore a strongly non linear system with large parameter m is transformed into a small parameter system with respect to a this method is suitable for the system with even non linearities as well as with odd non linearities all numerical results computed are in pretty good and displayed graphysically this method is suitable for the system with even non linearities as well as with odd non linearities all numerical results computed are in pretty good and displayed graphysically

a characterization theorem for semi classical orthogonal polynomials on non uniform lattices abstract it is proved a characterization theorem for semi classical orthogonal polynomials on non uniform lattices that states the equivalence between the pearson equation for the weight and some systems involving the orthogonal polynomials as well as the functions of the second kind as a consequence it is deduced the analogue of the so called compatibility conditions in the ladder operator scheme the classical orthogonal polynomials on non uniform lattices are then recovered under such compatibility conditions through a closed formula for the recurrence relation coefficients

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

on the stability in weak topology of the set of global solutions to the navier stokes equations let x be a suitable function space and let mathcal g subset x be the set of divergence free vector fields generating a global smooth solution to the incompressible homogeneous three dimensional navier stokes equations we prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of mathcal g belongs to mathcal g if n is large enough provided the convergence holds anisotropically in frequency space typically this excludes self similar type convergence anisotropy appears as an important qualitative feature in the analysis of the navier stokes equations it is also shown that initial data which do not belong to mathcal g hence which produce a solution blowing up in finite time cannot have a strong anisotropy in their frequency support

non uniform painless decompositions for anisotropic besov and triebel lizorkin spaces abstract in this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the lebesgue spaces l p r d 1 p the novelty and difficulty of this construction is that we allow for non lattice translations we prove that for an arbitrary expansive matrix a and any set satisfying a certain spreadness condition but otherwise irregular there exists a smooth window whose translations along the elements of and dilations by powers of a provide an atomic decomposition for the whole range of the anisotropic triebel lizorkin spaces the generating window can be either chosen to be bandlimited or to have compact support to derive these results we start with a known general painless construction that has recently appeared in the literature we show that this construction extends to besov and triebel lizorkin spaces by providing adequate dual systems

littlewood paley characterization and duality of weighted anisotropic product hardy spaces the authors study anisotropic product hardy spaces hwp a associated with a pair a a1 a2 of expansive dilations and a class of product muckenhoupt weights a a on rn rm this article is a continuation of their earlier work in bownik et al 2010 8 the authors establish the littlewood paley g function characterization and the transform characterization of hwp a 0 p 1 the authors also introduce the weighted anisotropic product campanato space lp w a and establish its transform characterization as an application the authors identify the dual space of hwp a with lp w a the results of this article improve the existing results for weighted product hardy spaces on r r and are new even in the weighted isotropic setting

the factorization method for the askey wilson polynomials a special infeld hull factorization is given for the askey wilson second order q difference operator it is then shown how to deduce a generalization of the corresponding askey wilson polynomials c 1999 elsevier science b v all rights reserved

duality and interpolation of anisotropic triebel lizorkin spaces we study properties of anisotropic triebel lizorkin spaces associated with general expansive dilations and doubling measures on mathbb r n using wavelet transforms this paper is a continuation of bownik in j geom anal 2007 to appear trans am math soc 358 1469 1510 2006 where we generalized the isotropic methods of dyadic varphi transforms of frazier and jawerth j funct anal 93 34 170 1990 to non isotropic settings by working at the level of sequence spaces we identify the duals of anisotropic triebel lizorkin spaces we also obtain several real and complex interpolation results for these spaces

bc_n symmetric abelian functions we construct a family of bc n symmetric biorthogonal abelian functions generalizing koornwinder s orthogonal polynomials and prove a number of their properties most notably analogues of macdonald s conjectures the construction is based on a direct construction for a special case generalizing okounkov s interpolation polynomials we show that these interpolation functions satisfy a collection of generalized hypergeometric identities including new multivariate elliptic analogues of jackson s summation and bailey s transformation

fourth order differential equations for numerator polynomials the author gives explicitly the fourth order differential equation satisfied by the numerator polynomials associated polynomials of the classical orthogonal polynomials the coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau and their derivatives defined via the relation sigma rho te this differential equation is therefore valid for the jacobi laguerre hermite and bessel associated polynomials

new besov type spaces and triebel lizorkin type spaces including q spaces let s tau in mathbb r and q in 0 infty we introduce besov type spaces dot b s tau p q mathbb r n for p in 0 infty and triebel lizorkin type spaces dot f s tau p q mathbb r n rm for p in 0 infty which unify and generalize the besov spaces triebel lizorkin spaces and q spaces we then establish the varphi transform characterization of these new spaces in the sense of frazier and jawerth using the varphi transform characterization of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n we obtain their embedding and lifting properties moreover for appropriate we also establish the smooth atomic and molecular decomposition characterizations of dot b s tau p q mathbb r n rm and dot f s tau p q mathbb r n for s in mathbb r p in 1 infty q in 1 infty and tau in 0 frac 1 max p q via the hausdorff capacity we introduce certain hardy hausdorff spaces b dot h s tau p q mathbb r n and prove that the dual space of b dot h s tau p q mathbb r n is just dot b s tau p q mathbb r n where t denotes the conjugate index of t in 1 infty

anisotropic variable hardy lorentz spaces and their real interpolation abstract let p r n 0 be a variable exponent function satisfying the globally log holder continuous condition q 0 and a be a general expansive matrix on r n in this article the authors first introduce the anisotropic variable hardy lorentz space h a p q r n associated with a via the radial grand maximal function and then establish its radial or non tangential maximal function characterizations moreover the authors also obtain characterizations of h a p q r n respectively in terms of the atom and the lusin area function as an application the authors prove that the anisotropic variable hardy lorentz space h a p q r n serves as the intermediate space between the anisotropic variable hardy space h a p r n and the space l r n via the real interpolation this together with a special case of the real interpolation theorem of h kempka and j vybiral on the variable lorentz space further implies the coincidence between h a p q r n and the variable lorentz space l p q r n when ess inf x r n p x 1

an analytical technique for solving a class of strongly nonlinear conservative systems abstract recently an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms usually a set of nonlinear algebraic equations is solved in this method however analytical solutions of these algebraic equations are not always possible especially in the case of a large oscillation previously such algebraic equations for the duffing equation were solved in powers of a small parameter but the solutions measure desired results when the amplitude is an order of 1 in this article different parameters of the same nonlinear problems are found for which the power series produces desired results even for the large oscillation moreover two or three terms of this power series solution measure a good result when the amplitude is an order of 1 besides these a suitable truncation formula is found in which the solution measures better results than existing solutions the method is mainly illustrated by the duffing oscillator but it is also useful for many other nonlinear problems

a generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators abstract an extended iteration method for calculating the periodic solutions of nonlinear oscillator equations is given the procedure is illustrated by applying it to two truly nonlinear oscillator differential equations

comments on investigation of the properties of the period for the nonlinear oscillator x 1 x 2 x 0 this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp006 007

vector orthogonal relations vector qd algorithm vector pade approximants to a function f f 1 f d from to d have been defined uniquely without any auxiliary choice than the degrees of the numerator and the denominator the same for all the components f i as in the scalar case 1 5 the denominators are associated to polynomials p r s which are given by vector orthogonal properties r and which satisfy for each s recurrence relations of order d 1 i e with d 2 terms called relations d we study here consequences of r and d first we prove an algorithm similar to the generalized mna al gorithm then we define a vector qd algorithm which links two diagonals p r s r and p r s 1 r conversely if a family p r s r 0 verifying d is given it is possible to build p r s r 0 s 0 and d linear functionals c a a 1 d such that p r 0 p r and p r s verify the orthogonal relations r with respect to the c a

asymptotic representations of the period for the nonlinear oscillator this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp006 007

function spaces of variable smoothness and integrability abstract in this article we introduce triebel lizorkin spaces with variable smoothness and integrability our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years vector valued maximal inequalities do not work in the generality which we pursue and an alternate approach is thus developed using it we derive molecular and atomic decomposition results and show that our space is well defined i e independent of the choice of basis functions as in the classical case a unified scale of spaces permits clearer results in cases where smoothness and integrability interact such as sobolev embedding and trace theorems as an application of our decomposition we prove optimal trace theorem in the variable indices case

orthogonal polynomials on systems of non uniform lattices from compatibility conditions abstract we deduce difference equations in the matrix form for laguerre hahn orthogonal polynomials on systems of non uniform lattices the so called compatibility conditions involving the transfer matrices as a consequence we obtain closed form expressions for the recurrence relation coefficients of the laguerre hahn polynomials of class zero

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

spectral theory of some degenerate elliptic operators with local singularities abstract this paper is based on our previous results haroske and skrzypczak 2008 23 haroske and skrzypczak in press 25 on compact embeddings of muckenhoupt weighted function spaces of besov and triebel lizorkin type with example weights of polynomial growth near infinity and near some local singularity our approach also extends haroske and triebel 1994 21 in various ways we obtain eigenvalue estimates of degenerate pseudodifferential operators of type b 2 p x d b 1 where b i l r i r n w i w i a i 1 2 and p x d 1 0 0 finally we deal with the negative spectrum of some operator h a v for where the potential v may have singularities in terms of muckenhoupt weights and a is a positive elliptic pseudodifferential operator of order 0 self adjoint in l 2 r n this part essentially relies on the birman schwinger principle we conclude this paper with a number of examples also comparing our results with preceding ones

characterizations of besov type and triebel lizorkin type spaces via maximal functions and local means abstract let s r in this paper the authors first establish the maximal function characterizations of the besov type space b p q s r n with p q 0 and 0 the triebel lizorkin type space f p q s r n with p 0 q 0 and 0 the besov hausdorff space b h p q s r n with p 1 q 1 and 0 1 max p q and the triebel lizorkin hausdorff space f h p q s r n with p q 1 and 0 1 max p q where t denotes the conjugate index of t 1 using this characterization the authors further obtain the local mean characterizations of these function spaces via functions satisfying the tauberian condition and establish a fourier multiplier theorem on these spaces all these results generalize the existing classical results on besov and triebel lizorkin spaces by taking 0 and are also new even for q spaces and hardy hausdorff spaces

investigation of the properties of the period for the nonlinear oscillator x 1 x 2 x 0 abstract the mathematical properties of a nonlinear oscillator having the square of the angular frequency depend quadratically on the velocity is presented the major conclusion is that the exact period t a is well defined for all values of the amplitude where the initial conditions are taken to be x 0 a and x 0 0 an approximate expression is derived for t a

co recursive orthogonal polynomials and fourth order differential equations abstract we give explicitly the 4th order differential equation satisfied by the co recursive polynomials of the classical orthogonal polynomials we emphasize the legendre and bessel cases for which the co recursive differential equation coincide with the equation satisfied by the numerator polynomial differential equation and we propose new conjectures

the structure relation for askey wilson polynomials an explicit structure relation for askey wilson polynomials is given this involves a divided q difference operator which is skew symmetric with respect to the askey wilson inner product and which sends polynomials of degree n to polynomials of degree n 1 by specialization of parameters and by taking limits similar structure relations as well as lowering and raising relations can be obtained for other families in the q askey scheme and the askey scheme this is explicitly discussed for jacobi polynomials continuous q jacobi polynomials continuous q ultraspherical polynomials and for big q jacobi polynomials an already known structure relation for this last family can be obtained from the new structure relation by using the three term recurrence relation and the second order q difference formula the results are also put in the framework of a more general theory their relationship with earlier work by zhedanov and bangerezako is discussed there is also a connection with the string equation in discrete matrix models and with the sklyanin algebra

on the q polynomials a distributed study in this paper we present a uni1ed distributional study of the classical discrete q polynomials in the hahn s sense from the distributional q pearson equation we will deduce many of their properties such as the three term recurrence relations structure relations etc also several characterizations of such q polynomials are presented c

a distributional study of discrete classical orthogonal polynomials abstract for the sequences of discrete classical orthogonal polynomials charlier meixner hahn we can find a functional u which satisfies the difference distributional equation u u where and are polynomials of degrees 2 and 1 respectively from this it follows that these polynomials are solutions of a second order difference equation also they can be represented by a rodrigues type formula the sequence of difference polynomials derived from them constitutes an orthogonal polynomial sequence their weight functions satisfy a pearson type difference equation a structure relation pn 1 anpn 2 bnpn 1 cnpn also holds

the factorization of a q difference equation for continuous q hermite polynomials we argue that a customary q difference equation for the continuous qhermite polynomials hn x q can be written in the factorized form as d q 2 1hn x q q n 1 hn x q where d q is some explicitly known q difference operator this means that the polynomials hn x q are in fact governed by the q difference equation d q hn x q q n 2 hn x q which is simpler than the conventional one it is shown that a similar factorization holds for the continuous q 1 hermite polynomials hn x q

new molecular characterizations of anisotropic musielak orlicz hardy spaces and their applications abstract let r n 0 0 be an anisotropic growth function and a a general expansive matrix on r n let h a r n be the anisotropic musielak orlicz hardy space associated with a defined via the non tangential grand maximal function in this article the authors establish a new molecular characterization of h a r n via its known atomic characterizations which positively answer a question mentioned by li et al 2016 34 as an application the authors obtain a criterion on the boundedness of linear operators on h a r n which further implies the boundedness of integral anisotropic calderon zygmund operators on h a r n in addition the boundedness of these operators from h a r n to the musielak orlicz space l r n is also presented

structure relations for monic orthogonal polynomials in two discrete variables in this paper extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given by using an appropriate vector matrix notation three term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type structure relations difference representations as well as lowering and raising operators are obtained finally expressions for all matrix coefficients appearing in these finite type relations are explicitly presented for a finite set of hahn and kravchuk orthogonal polynomials

dunkl shift operators and bannai ito polynomials abstract we consider the most general dunkl shift operator l with the following properties i l is of first order in the shift operator and involves reflections ii l preserves the space of polynomials of a given degree iii l is potentially self adjoint we show that under these conditions the operator l has eigenfunctions which coincide with the bannai ito polynomials we construct a polynomial basis which is lower triangular and two diagonal with respect to the action of the operator l this allows to express the bi polynomials explicitly we also present an anti commutator aw 3 algebra corresponding to this operator from the representations of this algebra we derive the structure and recurrence relations of the bi polynomials we introduce new orthogonal polynomials referred to as the complementary bi polynomials as an alternative q 1 limit of the askey wilson polynomials these complementary bi polynomials lead to a new explicit expression for the bi polynomials in terms of the ordinary wilson polynomials