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

a laplace dunkl equation on s 2 and the bannai ito algebra the dirac dunkl operator on the two sphere associated to the mathbb z 2 3 reflection group is considered its symmetries are found and are shown to generate the bannai ito algebra representations of the bannai ito algebra are constructed using ladder operators eigenfunctions of the spherical dirac dunkl operator are obtained using a cauchy kovalevskaia extension theorem these eigenfunctions which correspond to dunkl monogenics are seen to support finite dimensional irreducible representations of the bannai ito algebra

the heun racah and heun bannai ito algebras the heun racah and heun bannai ito algebras are introduced specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic heun construct to the bispectral operators of the racah and bannai ito polynomials the study supplements the results on the heun askey wilson algebra and completes the description of the heun algebras associated with the polynomial families at the top of the askey scheme its q analog and the bannai ito one

a missing family of classical orthogonal polynomials we study a family of classical orthogonal polynomials which satisfy apart from a three term recurrence relation an eigenvalue problem with a differential operator of dunkl type these polynomials can be obtained from the little q jacobi polynomials in the limit q 1 we also show that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1

global existence and asymptotic behavior of classical solutions to a fractional logistic keller segel system abstract in this paper we consider a fractional parabolic elliptic keller segel system with a logistic source on r n first we establish the regularity of weak solutions of the fractional parabolic equation using blow up arguments combined with liouville type theorems next by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions moreover the global existence and boundedness of classical solutions for given initial data are obtained under some conditions finally we show the asymptotic behavior of the global solutions with strictly positive initial data

symmetric functions and bn invariant spherical harmonics the wavefunctions of a quantum isotropic harmonic oscillator modified by reflecting barriers at the coordinate planes in n dimensional space can be expressed in terms of certain generalized spherical harmonics these are associated with a product type weight function on the sphere their analysis is carried out by means of differential difference operators the symmetries of this system involve the weyl group of type b generated by permutations and changes of sign of the coordinates a new basis for symmetric functions as well as an explicit transition matrix to the monomial basis is constructed this basis leads to a basis for invariant spherical harmonics the determinant of the gram matrix for the basis in the natural inner product over the sphere is evaluated when the underlying parameter is specialized to zero the basis consists of ordinary spherical harmonics with cube group symmetry as used for wavefunctions of electrons in crystals the harmonic oscillator can also be considered as a degenerate interaction free spin calogero model

convolution identities for dunkl orthogonal polynomials from the osp 1 2 lie superalgebra new convolution identities for orthogonal polynomials belonging to the q 1 analog of the askey scheme are obtained specialization of the chihara polynomials will play a central role as the eigenfunctions of a special element of the lie superalgebra osp 1 2 in the positive discrete series representation using the clebsch gordan coefficients a convolution identity for the specialized chihara the dual 1 hahn and the big 1 jacobi polynomials are found using the racah coefficients a convolution identity for the big 1 jacobi and the bannai ito polynomials is found finally these results are applied to construct a bilinear generating function for the big 1 jacobi polynomials new convolution identities for orthogonal polynomials belonging to the q 1 analog of the askey scheme are obtained specialization of the chihara polynomials will play a central role as the eigenfunctions of a special element of the lie superalgebra osp 1 2 in the positive discrete series representation using the clebsch gordan coefficients a convolution identity for the specialized chihara the dual 1 hahn and the big 1 jacobi polynomials are found using the racah coefficients a convolution identity for the big 1 jacobi and the bannai ito polynomials is found finally these results are applied to construct a bilinear generating function for the big 1 jacobi polynomials

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

an algebraic approach to shape invariance with reflections we consider a realization of supersymmetric shape invariant quantum systems where generalized ladder operators are constructed through the association of the supersymmetric partner operators the parameter potential translation operator and the reflection operator the basic algebraic relations as well as the system eigenstates and energy eigenvalues are obtained as an illustration we apply the generalized formalism for two important shape invariant potential systems the scarf hyperbolic and the morse potentials

coefficient ranges necessary for stability of 2 d polynomials it is shown that for the stability of a 2 d polynomial it is necessary that its coefficient values are within certain ranges these ranges are determined by extending the results obtained in 3 for the 1 d case by simply comparing the coefficient values with the ranges necessary for stability instability can be concluded in some cases in these cases any computation for 2 d or n d stability testing can be avoided

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

the hahn superalgebra and supersymmetric dunkl oscillator models a supersymmetric extension of the hahn algebra is introduced this quadratic superalgebra which we call the hahn superalgebra is constructed using the realization provided by the dunkl oscillator model in the plane whose hamiltonian involves reflection operators in this realization the reflections act as grading operators and the odd generators are part of the schwinger dunkl algebra which is a two parameter extension of the bosonic construction the even part of the algebra is built from bilinears in the odd generators and satisfies the hahn algebra supplemented with involutions a family of supersymmetric dunkl oscillator models in n dimensions is also considered the hamiltonians of these supersymmetric models differ from ordinary dunkl oscillators by pure reflection terms in two dimensions the supersymmetric dunkl oscillator is seen to have the even part of the hahn superalgebra as the invariance algebra

dunkl operators and a family of realizations of osp 1 2 in this paper a family of radial deformations of the realization of the lie superalgebra osp 1 vertical bar 2 in the theory of dunkl operators is obtained this leads to a dirac operator depending on 3 parameters several function theoretical aspects of this operator are studied such as the associated measure the related laguerre polynomials and the related fourier transform for special values of the parameters it is possible to construct the kernel of the fourier transform explicitly as well as the related intertwining operator

the algebra of dual 1 hahn polynomials and the clebsch gordan problem of sl 1 2 the algebra h of the dual 1 hahn polynomials is derived and shown to arise in the clebsch gordan problem of sl 1 2 the dual 1 hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q 1 limit of the dual q hahn polynomials the hopf algebra sl 1 2 has four generators including an involution it is also a q 1 limit of the quantum algebra slq 2 and furthermore the dynamical algebra of the parabose oscillator the algebra h a two parameter generalization of u 2 with an involution as additional generator is first derived from the recurrence relation of the 1 hahn polynomials it is then shown that h can be realized in terms of the generators of two added sl 1 2 algebras so that the clebsch gordan coefficients of sl 1 2 are dual 1 hahn polynomials an irreducible representation of h involving five diagonal matrices and connected to the difference equation of the dual 1 hahn polynomials is constructed

a higher rank racah algebra and the mathbb z _2 n laplace dunkl operator a higher rank generalization of the rank one racah algebra is obtained as the symmetry algebra of the laplace dunkl operator associated to the mathbb z 2 n root system this algebra is also the invariance algebra of the generic superintegrable model on the n sphere bases of dunkl harmonics are constructed explicitly using a cauchy kovalevskaia theorem these bases consist of joint eigenfunctions of labelling abelian subalgebras of the higher rank racah algebra a method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented

a superintegrable model with reflections on s n 1 and the higher rank bannai ito algebra a quantum superintegrable model with reflections on the n 1 sphere is presented its symmetry algebra is identified with the higher rank generalization of the bannai ito algebra it is shown that the hamiltonian of the system can be constructed from the tensor product of n representations of the superalgebra osp 1 2 and that the superintegrability is naturally understood in that setting the separated solutions are obtained through the fischer decomposition and a cauchy kovalevskaia extension theorem

eigenproblems for tridiagonal 2 toeplitz matrices and quadratic polynomial mappings abstract given a system of monic orthogonal polynomials mops p n x n 0 we characterize all the sequences of monic orthogonal polynomials q n x n 0 such that q 1 x x b q 2 n x p n 2 x n 0 1 2 where 2 is a fixed polynomial of degree exactly 2 and b is a fixed complex number with an appropriate choice of the mops p n x n 0 our results enables us to solve the eigenproblem of a tridiagonal 2 toeplitz matrix giving an alternative proof to a recent result by m j c gover we also find the relations between the jacobi matrices corresponding to the mops p n x n 0 and q n x n 0 finally we show that if p n x n 0 is a semiclassical orthogonal polynomial sequence then so is q n x n 0 and in particular we analyze the classical case in detail

bivariate bannai ito polynomials a two variable extension of bannai ito polynomials is presented they are obtained via q 1 limits of the bivariate q racah and askey wilson orthogonal polynomials introduced by gasper and rahman their orthogonality relation is obtained these new polynomials are also shown to be multispectral two dunkl shift operators are seen to be diagonalized by the bivariate bannai ito polynomials and 3 and 9 term recurrence relations are provided a two variable extension of bannai ito polynomials is presented they are obtained via q 1 limits of the bivariate q racah and askey wilson orthogonal polynomials introduced by gasper and rahman their orthogonality relation is obtained these new polynomials are also shown to be multispectral two dunkl shift operators are seen to be diagonalized by the bivariate bannai ito polynomials and 3 and 9 term recurrence relations are provided

the q onsager algebra and multivariable q special functions two sets of mutually commuting q difference operators x i and y j i j 1 n such that x i and y i generate a homomorphic image of the q onsager algebra for each i are introduced the common polynomial eigenfunctions of each set are found to be entangled product of elementary pochhammer functions in n variables and n 3 parameters under certain conditions on the parameters they form two dual bases of polynomials in n variables the action of each operator with respect to its dual basis is block tridiagonal the overlap coefficients between the two dual bases are expressed as entangled products of q racah polynomials and satisfy an orthogonality relation the overlap coefficients between either one of these bases and the multivariable monomial basis are also considered one obtains in this case entangled products of dual q krawtchouk polynomials finally the split basis in which the two families of operators act as block bidiagonal matrices is also provided

h norm and invariant manifolds of systems with state delays abstract the problem of finding bounds on the h norm of systems with a finite number of point delays and distributed delay is considered sufficient conditions for the system to possess an h norm which is less or equal to a prescribed bound are obtained in terms of riccati partial differential equations rpde s we show that the existence of a solution to the rpde s is equivalent to the existence of a stable manifold of the associated hamiltonian system for small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the hamiltonian system this leads to an algebraic finite dimensional criterion for systems with small delays

long time dynamics of forced critical sqg we prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi geostrophic equation sqg and prove that it has finite fractal box counting dimension in order to do so we give a new proof of global regularity for critical sqg the main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional laplacian which is used to bootstrap the regularity directly from l to c without the use of de giorgi techniques we prove that for large time the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force which is assumed to belong merely to l h1 using the fact that the solution is bounded independently of the initial data after a transient time in spaces conferring enough regularity we prove the existence of a compact absorbing set for the dynamics in h1 obtain the compactness of the linearization and the continuous differentiability of the solution map we then prove exponential decay of high yet finite dimensional volume elements in h1 along solution trajectories and use this property to bound the dimension of the global attractor

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

generating functions for the mathfrak osp 1 2 clebsch gordan coefficients generating functions for clebsch gordan coefficients of are derived these coefficients are expressed as limits of the dual q hahn polynomials the generating functions are obtained using two different approaches respectively based on the coherent state representation and the position representation of

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

generalized gegenbauer orthogonal polynomials abstract in this paper we explore a specific semi classical orthogonal sequence namely the generalized gegenbauer orthogonal polynomials gg which appear in many applications such as the weighted l p mean convergence of hermite fejer interpolation or the chain of harmonic oscillators in the absence of externally applied forces first we trace back the genesis of gg underlining its links with the jacobi orthogonal polynomials second we establish a differential difference relation and the second order differential equation satisfied by this sequence we end by giving the fourth order differential equation satisfied by the association of arbitrary order of the gg

global existence and asymptotic stability of the fractional chemotaxis fluid system in r3 abstract in this paper we consider a model arising from biology consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in r 3 we establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument moreover the decay rates of the solutions and their higher order spatial derivatives are established towards the equilibrium by introducing the negative sobolev space h s 0 s 3 2

algebraic heun operator and band time limiting we introduce the algebraic heun operator associated to any bispectral pair of operators this operator can be presented as a generic bilinear combination of these bispectral operators we show that the resulting operators are natural generalizations of the ordinary heun operator this leads to a simple construction of the operators commuting with the projection operators in problems of band time limiting and it gives a way to adapt a construction first used by perline in a purely finite setup to general situations we also extend this algebraic construction to cover some purely finite cases which lie beyond this scheme in particular for the anti krawtchouk polynomials in the latter case the commuting operator is represented by a pentadiagonal matrix which can be constructed explicitly as a polynomial of degree 4 of the pair of bispectral operators

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

suppression of blow up by a logistic source in 2d keller segel system with fractional dissipation we consider a two dimensional parabolic elliptic keller segel equation with a logistic forcing and a fractional diffusion of order we obtain existence of global in time regular solution for arbitrary initial data with no size restrictions and c 2 where c 0 2 depends on the equation s parameters for an even wider range of s we prove existence of global in time weak solution for general initial data

some generalized jacobi polynomials abstract we give explicitly the recurrence coefficients in the three term recurrence relation of some generalized jacobi polynomials defined by the positive weight p x 1 x 2 1 x p on 1 1 the case p 0 can be found in chihara s book the case p 1 is treated by the first author and we consider here the cases p 2 3 4 these recurrence coefficients can be used to prove that their corresponding polynomials are random walk and then are of interest in birth and death process 1 2 for instance the cases p 1 2 are treated

global solutions for a supercritical drift diffusion equation abstract we study the global existence of solutions to a one dimensional drift diffusion equation with logistic term generalizing the classical parabolic elliptic keller segel aggregation equation arising in mathematical biology in particular we prove that there exists a global weak solution if the order of the fractional diffusion 1 c 1 2 where c 1 0 is an explicit constant depending on the physical parameters present in the problem chemosensitivity and strength of logistic damping furthermore in the range 1 c 2 2 with 0 c 2 c 1 the solution is globally smooth let us emphasize that when 1 the diffusion is in the supercritical regime

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

modular leonard triples abstract let k denote a field and let v denote a vector space over k of finite positive dimension an ordered triple a a a of linear operators on v is said to be a leonard triple whenever for each b a a a there exists a basis of v with respect to which the matrix representing b is diagonal and the matrices representing the other two operators are irreducible tridiagonal a leonard triple a a a is said to be modular whenever for each b a a a there exists an antiautomorphism of end v which fixes b and swaps the other two operators we classify the modular leonard triples up to isomorphism

a limit q 1 for the big q jacobi polynomials we study a new family of classical orthogonal polynomials here called big 1 jacobi polynomials which satisfy apart from a 3 term recur rence relation an eigenvalue problem with differential operators of dunkl type these polynomials can be obtained from the big q jacobi polynomials in the limit q 1 an explicit expression of these polynomials in terms of gauss hypergeometric functions is found the big 1 jacobi polynomials are orthogo nal on the union of two symmetric intervals of the real axis we show that the big 1 jacobi polynomials can be obtained from the bannai ito polynomials when the orthogonality support is extended to an infinite number of points we further indicate that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1 the novelty lies in the fact that l is a differential difference operator of special type namely l is a linear operator which is of first order in the derivative operator x and contains also the reflection operator r which acts as rf x f x roughly speaking one can say that l belongs to the class of dunkl operators 8 which contain both the operators x and r nevertheless the operator l differs from the standard dunkl operators in a fundamental way indeed l preserves the linear space of polynomials of any given maximal degree this basic property allows to construct a complete system of polynomials pn x n 0 1 2 as eigenfunctions of the operator l guided by the q 1 limit of the little q jacobi polynomials we derived in 20 an explicit expression of the polynomials pn x in terms of gauss hypergeometric functions we also found explicitly the recurrence coefficients and s that the polynomials pn x are orthogonal on the interval 1 1 with a weight function related to the weight function of the generalized jacobi polynomials 7 we also proved that they admit the dunkl classical property 5 and further demonstrated that the operator l together with the multiplication operator x form a special case of the askey wilson algebra aw 3 21 corresponding to the parameter q 1

global existence for some transport equations with nonlocal velocity abstract in this paper we study transport equations with nonlocal velocity fields with rough initial data we address the global existence of weak solutions of a one dimensional model of the surface quasi geostrophic equation and the incompressible porous media equation and one dimensional and n dimensional models of the dissipative quasi geostrophic equations and the dissipative incompressible porous media equation

darboux transformation and perturbation of linear functionals abstract let l be a quasi definite linear functional defined on the linear space of polynomials with real coefficients in the literature three canonical transformations of this functional are studied xl l c x and 1 x l c x where x denotes the linear functional x xk k 0 and k 0 is the kronecker symbol let us consider the sequence of monic polynomials orthogonal with respect to l this sequence satisfies a three term recurrence relation whose coefficients are the entries of the so called monic jacobi matrix in this paper we show how to find the monic jacobi matrix associated with the three canonical perturbations in terms of the monic jacobi matrix associated with l the main tools are darboux transformations in the case that the lu factorization of the monic jacobi matrix associated with xl does not exist and darboux transformation does not work we show how to obtain the monic jacobi matrix associated with x 2 l as a limit case we also study perturbations of the functional l that are obtained by combining the canonical cases finally we present explicit algebraic relations between the polynomials orthogonal with respect to l and orthogonal with respect to the perturbed functionals

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

the dunkl oscillator in the plane ii representations of the symmetry algebra the superintegrability wavefunctions and overlap coefficients of the dunkl oscillator model in the plane were considered in the first part here finite dimensional representations of the symmetry algebra of the system called the schwinger dunkl algebra sd 2 are investigated the algebra sd 2 has six generators including two involutions and a central element and can be seen as a deformation of the lie algebra mathfrak u 2 two of the symmetry generators j 3 and j 2 are respectively associated to the separation of variables in cartesian and polar coordinates using the parabosonic creation annihilation operators two bases for the representations of sd 2 the cartesian and circular bases are constructed in the cartesian basis the operator j 3 is diagonal and the operator j 2 acts in a tridiagonal fashion in the circular basis the operator j 2 is block upper triangular with all blocks 2 2 and the operator j 3 acts in a tridiagonal fashion the expansion coefficients between the two bases are given by the krawtchouk polynomials in the general case the eigenvectors of j 2 in the circular basis are generated by the heun polynomials and their components are expressed in terms of the para krawtchouk polynomials in the fully isotropic case the eigenvectors of j 2 are generated by little 1 jacobi or ordinary jacobi polynomials the basis in which the operator j 2 is diagonal is considered in this basis the defining relations of the schwinger dunkl algebra imply that j 3 acts in a block tridiagonal fashion with all blocks 2 2 the matrix elements of j 3 in this basis are given explicitly

refined blowup criteria and nonsymmetric blowup of an aggregation equation abstract we consider an aggregation equation in r d d 2 with fractional dissipation u t u k u u where 0 0 1 and k x e x we prove a refined blowup criteria by which the global existence of solutions is controlled by its l x q norm for any d d 1 q we prove the finite time blowup of solutions for a general class of nonsymmetric initial data the argument presented works for both the inviscid case 0 and the supercritical case 0 and 0 1 additionally we present new proofs of blowup which does not use free energy arguments

global existence and asymptotic behavior of the fractional chemotaxis system with signal dependent sensitivity abstract in this paper we study a fractional chemotaxis model with signal dependent sensitivity and the source term which involves two parabolic equations with fractional diffusions we develop a suitable mathematical framework for a unified treatment of the existence and decay estimates based on an appropriate functional space we show the existence uniqueness and temporal decay of the global classical solutions to the problem simultaneously under the smallness initial assumptions

dunkl supersymmetric orthogonal functions associated with classical orthogonal polynomials we consider the eigenvalue problem associated with the dunkl type differential operator in which the reflection operator r is involved l dx r v x v x v x in the context of supersymmetric quantum mechanical models by solving this eigenvalue prob lem with the help of known exactly solvable potentials we construct several classes of func tions satisfying certain orthogonality relations we call them the dunkl supersymmetric dunkl susy orthogonal functions these functions can be expressed in terms of the classical orthog onal polynomials cops the key feature of these functions is that they appear by pairs i e qn x and qn x are both the eigenfunctions of l a general formulation of the dunkl susy orthogonal polynomials is also presented

exact slow fast decomposition of the nonlinear singularly perturbed optimal control problem abstract we study the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system which is nonlinear in both the slow and the fast variables it is known that the optimal controller for such problem can be designed by finding a special invariant manifold of the corresponding hamiltonian system we obtain exact slow fast decomposition of the hamiltonian system and of the special invariant manifold into the slow and the fast ones on the basis of this decomposition we construct high order asymptotic approximations of the optimal state feedback and optimal trajectory

leonard triples of q racah type and their pseudo intertwiners abstract let f denote a field and let v denote a vector space over f with finite positive dimension pick a nonzero q f such that q 4 1 and let a b c denote a leonard triple on v that has q racah type we show that there exist invertible w w w in end v such that i a commutes with w and w 1 b w c ii b commutes with w and w 1 c w a iii c commutes with w and w 1 a w b moreover each of w w w is unique up to multiplication by a nonzero scalar in f we show that the three elements w w w w w w mutually commute and their product is a scalar multiple of the identity a number of related results are obtained we call w w w the pseudo intertwiners for a b c

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

revisiting the askey wilson algebra with the universal r matrix of uq su 2 a description of the embedding of a centrally extended askey wilson algebra aw 3 in uq sl2 3 is given in terms of the universal r matrix of uq sl2 the generators of the centralizer of uq sl2 in its three fold tensor product are naturally expressed through conjugations of casimir elements with r they are seen as the images of the generators of aw 3 under the embedding map by showing that they obey the aw 3 relations this is achieved by introducing a natural coaction also constructed with the help of the r matrix

spectral transformations self similar reductions and orthogonal polynomials we study spectral transformations in the theory of orthogonal polynomials which are similar to darboux transformations for the schr dinger equation linear transformations of the stieltjes function with coefficients that are rational in the argument are constructed as iterations of the christoffel and geronimus transformations we describe a characteristic property of semi classical orthogonal polynomials scop on the uniform and the exponential lattice namely that all these polynomials can be obtained through simple quasi periodic and q periodic self similar closures of the chain of linear spectral transformations in the self similar setting a characterization of the laguerre hahn polynomials on linear and q linear lattices is obtained by considering rational transformations of the stieltjes function generated by transitions to the associated polynomials

jordan algebras and orthogonal polynomials we illustrate how jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials the big 1 jacobi polynomials are eigenfunctions of a first order operator of dunkl type we consider an algebra that has this operator up to constants as one of its three generators and whose defining relations are given in terms of anticommutators it is a special case of the askey wilson algebra aw 3 we show how the structure and recurrence relations of the big 1 jacobi polynomials are obtained from the representations of this algebra we also present ladder operators for these polynomials and point out that the big 1 jacobi polynomials satisfy the hahn property with respect to a generalized dunkl operator

centralizers of the superalgebra mathfrak osp 1 2 o s p 1 2 the brauer algebra as a quotient of the bannai ito algebra

a bosonized formulation of the supersymmetric quantum mechanics for shape invariant potential systems we consider the minimal bosonization realization of supersymmetric shape invariant systems where generalized supercharge operators are constructed using the partner supersymmetric operators the parameter potential translation formalism and the reflection operator we obtain the solution of the eigenvalue equation and study the quantum dynamics of the supersymmetric system including terms in the hamiltonian which are constructed using the combination of the bosonized supercharge operators the connections between the bosonized supersymmetric formalism the bose fermi transformation and the generalization of the deformed heisenberg algebra are discussed as an illustration we apply the generalized formalism for the case of the trigonometric rosen morse potential

the z 2 n dirac dunkl operator and a higher rank bannai ito algebra abstract the kernel of the z 2 n dirac dunkl operator is examined the symmetry algebra a n of the associated dirac dunkl equation on s n 1 is determined and is seen to correspond to a higher rank generalization of the bannai ito algebra a basis for the polynomial null solutions of the dirac dunkl operator is constructed the basis elements are joint eigenfunctions of a maximal commutative subalgebra of a n and are given explicitly in terms of jacobi polynomials the symmetry algebra is shown to act irreducibly on this basis via raising lowering operators a scalar realization of a n is proposed

the higher rank q deformed bannai ito and askey wilson algebra the q deformed bannai ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra mathfrak osp q 1 vert 2 it turned out to be isomorphic to the askey wilson algebra in the present paper these results will be extended to higher rank the rank n 2 q bannai ito algebra mathcal a n q which by the established isomorphism also yields a higher rank version of the askey wilson algebra is constructed in the n fold tensor product of mathfrak osp q 1 vert 2 an explicit realization in terms of q shift operators and reflections is proposed which will be called the mathbb z 2 n q dirac dunkl model the algebra mathcal a n q is shown to arise as the symmetry algebra of the constructed mathbb z 2 n q dirac dunkl operator and to act irreducibly on modules of its polynomial null solutions an explicit basis for these modules is obtained using a q deformed mathbf ck extension and fischer decomposition

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

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

families of 2d superintegrable anisotropic dunkl oscillators and algebraic derivation of their spectrum we generalize the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach this is presented in the context of 1d systems admitting ladder operators satisfying a parabosonic algebra involving reflection operators and more generally c extended oscillator algebras with grading we apply the construction on twodimensional c oscillators we also introduce two new superintegrable hamiltonians that are the anisotropic dunkl and the singular dunkl oscillators integrals are constructed by extending the approach of daskaloyannis to include grading an algebraic derivation of the energy spectra of the two models is presented making use of finite dimensional unitary representations we show how the spectra divide into sectors and make comparisons with the physical case