a characterization of hermite polynomials we show that for any orthogonal polynomials p x o satisfying a spectral type differential equation of order n 2 n ln y x x y i x 2 y x i i pn x 0 must be essentially hermite polynomials if and only if the leading coefficient en x is a nonzero constant

orthogonal and extremal polynomials on several intervals abstract let a1 1 where is positive on el are characterised by orthogonality conditions and some new results are proved then the connection with polynomials orthogonal with respect to h where has an odd number of zeros in each interval a2j a2j 1 j 1 l 1 is shown and a full description of orthogonal polynomials having periodic respectively asymptotic periodic recurrence coefficients is given finally the asymptotic behaviour of polynomials orthogonal on several intervals is discussed

a matrix characterization for the dν semiclassical and dν coherent orthogonal polynomials abstract we present a new structure relation for the sequence of orthogonal polynomials associated with a d semiclassical linear functional of class s and then we use it to obtain a matrix characterization of the d semiclassical orthogonal polynomials in terms of the jacobi matrix associated with the multiplication operator in the basis of orthonormal polynomials and the nonsingular lower triangular matrix that represents the orthogonal polynomials with respect to some bases of polynomials we also provide a matrix characterization of d coherent pairs of linear functionals

the generalized bochner condition about classical orthogonal polynomials revisited we bring a new proof for showing that an orthogonal polynomial sequence is classical if and only if any of its polynomial fulfils a certain differential equation of order 2k for some k 1 so we build those differential equations explicitly if k 1 we get the bochner s characterization of classical polynomials with help of the formal computations made in mathematica we explicitly give those differential equations for k 1 2 and 3 for each family of the classical polynomials higher order differential equations can be obtained similarly

a general method for deriving some semi classical properties of perturbed second degree forms we present a new general method and the corresponding symbolic algorithm psdf for deriving some semi classical properties of perturbed second degree forms namely the stieltjes function the stieltjes equation the functional equation the class a structure relation and the second order linear differential equation we give new explicit results for some perturbed of order 3 of the second kind chebyshev polynomials

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings

orthogonal and l q extremal polynomials on inverse images of polynomial mappings abstract let t be a polynomial of degree n and let k be a compact set with c first it is shown if zero is a best approximation to f from p n on k with respect to the lq norm q 1 then zero is also a best approximation to f t on t 1 k with respect to the l q t norm where t arises from by the transformation t in particular t is the equilibrium measure on t 1 k if is the equilibrium measure on k for q i e the sup norm a corresponding result is presented in this way polynomials minimal on several intervals on lemniscates on equipotential lines of compact sets etc are obtained special attention is given to lq minimal polynomials on julia sets next based on asymptotic results of widom we show that the minimum deviation of polynomials orthogonal with respect to a positive measure on t 1 k behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour too some open problems are also given

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

a method to study the krall and q krall polynomials we describe a method allowing to obtain new results in studying the krall and q krall polynomials in particular we establish the criterion for orthogonal polynomials to be eigenfunctions of 2nth order q differential operator this criterion is an exact q analogue of corresponding krall s criterion

the classical laurent biorthogonal polynomials abstract an analogue of the hahn theorem for laurent biorthogonal polynomials lbp pn z is studied necessary and sufficient conditions criterion for derivatives p n z n 1 1 p n 1 z to be lbp are obtained this criterion can be written as a linear second order difference equation for the moments we find examples of such lbp which are different from the well known classical lbp proposed by hendriksen and van rossum the theory of spectral transformations of the lbp i e analogs of christoffel and geronimus transforms is developed these transformations are shown to be crucial in study of lbp belonging to the hahn class

a characterization of classical and semiclassical orthogonal polynomials from their dual polynomials we study properties of the dual orthogonal polynomials op introduced by de boor and saff and present a new characterization of the classical and semiclassical op

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

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

meixner polynomials of the second kind and quantum algebras representing su 1 1 we show how viennot s combinatorial theory of orthogonal polynomials may be used to generalize some recent results of sukumar and hodges hodges sukumar 2007 proc r soc a 463 2401 2414 doi 10 1098 rspa 2007 0001 1 sukumar hodges 2007 proc r soc a 463 2415 2427 doi 10 1098 rspa 2007 0003 2 on the matrix entries in powers of certain operators in a representation of su 1 1 our results link these calculations to finding the moments and inverse polynomial coefficients of certain laguerre polynomials and meixner polynomials of the second kind as an immediate consequence of results by koelink groenevelt and van der jeugt van der jeugt 1997 j math phys 38 2728 2740 doi 10 1063 1 531984 3 koelink van der jeugt 1998 siam j math anal 29 794 822 doi 10 1137 s003614109630673x 4 groenevelt koelink 2002 j phys a 35 65 85 doi 10 1088 0305 4470 35 1 306 5 for the related operators substitutions into essentially the same laguerre polynomials and meixner polynomials of the second kind may be used to express their eigenvectors our combinatorial approach explains and generalizes this coincidence 1 http dx doi org doi 10 1098 rspa 2007 0001 2 http dx doi org doi 10 1098 rspa 2007 0003 3 http dx doi org doi 10 1063 1 531984 4 http dx doi org doi 10 1137 s003614109630673x 5 http dx doi org doi 10 1088 0305 4470 35 1 306

the measure of the orthogonal polynomials related to fibonacci chains the periodic case the spectral measure for the two families of orthogonal polynomial systems related to periodic chains with n particle elementary unit and nearest neighbour harmonic interaction is computed using two different methods the interest is in the orthogonal polynomials related to fibonacci chains in the periodic approximation the relation of the measure to appropriately defined green s functions is established

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

spectral analysis of a two body problem with zero range perturbation abstract we consider a class of singular zero range perturbations of the hamiltonian of a quantum system composed by a test particle and a harmonic oscillator in dimension one two and three and we study its spectrum in fact we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the hamiltonian we also partially describe the positive spectrum and scattering properties of the hamiltonian

iterative solution of indefinite symmetric linear systems by methods using orthogonal polynomials over two disjoint intervals in the classical chebyshev algorithm for solving symmetric positive definite linear systems ax b the approximate solution is of the form x k x 0 s k a r 0 where r 0 b ax 0 is the initial residual and s k is a polynomial of degree k 1 which minimizes 1 lambda s k lambda infty where cdot infty is the uniform norm on some interval s containing the spectrum of a but not the origin if a is not positive definite the set s can be taken to be the union of two disjoint intervals a b and c d with b 0 c and a generalization of the chebyshev algorithm based on the minimax polynomial has been considered by several authors in this paper we propose a new technique based upon the least squares polynomial in the set s i e the polynomial t k which minimizes 1 lambda t k lambda w where cdot w is an l 2 norm with respect to a weight function w lambda defined on s with a suitable weight function w lambda the po

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

variations around classical orthogonal polynomials connected problems abstract we survey and prove the principal characterizations of the classical orthogonal polynomials with the method of the dual sequence two new characterizations are given connected problems are discussed

orthogonality properties of iterated polynomial mappings we consider a measure defined on a complex contour and its associated orthogonal polynomials the action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given iterating the transformation provides an invariant measure whose support is the julia set corresponding to the polynomial transformation it appears that up to a constant the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials which fulfill a three term recursion relation an algorithm is given to compute the coefficients of this recursion relation which can be interpreted as a linear extension of the iterative procedure

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

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

new results on the bochner condition about classical orthogonal polynomials the classical polynomials hermite laguerre bessel and jacobi are the only orthogonal polynomial sequences ops whose elements are eigenfunctions of the bochner second order differential operator f bochner 1929 3 in loureiro maroni and da rocha 2006 18 these polynomials were described as eigenfunctions of an even order differential operator fk with polynomial coefficients defined by a recursive relation here an explicit expression of fk for any positive integer k is given the main aim of this work is to explicitly establish sums relating any power of f with fk k 1 in other words to bring a pair of inverse relations between these two operators this goal is accomplished with the introduction of a new sequence of numbers the so called a modified stirling numbers which could be also called as bessel or jacobi stirling numbers depending on the context and the values of the complex parameter a

krall type polynomials via the heine formula with the heine multiple integral formula we study the effect of a special class of perturbations adding and to the weights on the laguerre and jacobi polynomials

on some n 1 symmetric linear functionals abstract many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle particular aspects are treated for quadratic or cubic decompositions in this paper we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality this technique is based on the decomposition with respect to the cyclic group of order n the main theorem is applied to the orthogonality on n 1 symmetric curves some particular cases are singled out these results may be useful in studying hermite pade approximations vector continued fractions and dynamical systems

an integral representation for the bessel form abstract we deal with integral representation problems of the bessel form suitable formulations are obtained but they are not proved for all values of the parameter generalizations to modified classical forms are possible

construction of differential operators having bochner krall orthogonal polynomials as eigenfunctions abstract suppose q n n 0 is a sequence of polynomials orthogonal with respect to the moment functional where is a classical moment functional jacobi laguerre hermite and is a point mass distribution with finite support in this paper we develop a new method for constructing a differential equation having q n n 0 as eigenfunctions

universality limits for generalized jacobi measures abstract in this paper universality limits are studied in connection with measures which exhibit power type singular behavior somewhere in their support we extend the results of lubinsky for jacobi measures supported on 1 1 to generalized jacobi measures supported on a compact subset of the real line where the singularity can be located in the interior or at an endpoint of the support the analysis is based upon the riemann hilbert method christoffel functions the polynomial inverse image method of totik and the normal family approach of lubinsky

on the inverse problem of the product of a semi classical form by a polynomial abstract a form linear functional u is called regular if there exists a sequence of polynomials pn n 0 deg pn n which is orthogonal with respect to u such a form is said to be semi classical if there exist polynomials and such that d u u 0 where d designs the derivative operator on certain regularity conditions the product of a semi classical form by a polynomial gives a semi classical form in this paper we consider the inverse problem given a semi classical form v find all regular forms u which satisfy the relation x2u v c we give the structure relation differential recurrence relation of the orthogonal polynomial sequence relatively to u an example is treated with a nonsymmetric form v

inverse images of polynomial mappings and polynomials orthogonal on them let j be a polynomial with complex coefficients first we study the inverse images of the real and imaginary axes under a polynomial mapping j in detail then for an arbitrary polynomial and a sequence pn of orthogonal polynomials the orthogonality behaviour of the sequence of polynomials pn j n n is investigated in particular necessary and sufficient conditions are given such that pn j n n is a subsequence of polynomials orthogonal with respect to a positive measure supported on a compact subset of the real line

a remarkable spectral feature of the schrodinger hamiltonian of the harmonic oscillator perturbed by an attractive δ interaction centred at the origin double degeneracy and level crossing we rigorously define the self adjoint hamiltonian of the harmonic oscillator perturbed by an attractive interaction of strength centred at 0 the bottom of the confining parabolic potential by explicitly providing its resolvent our approach is based on a coupling constant renormalization related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the interaction in two and three dimensions the way the interaction enters in our hamiltonian corresponds to the one originally discussed for the free hamiltonian instead of the harmonic oscillator one by p s ba it should not be confused with the potential perturbation of the harmonic oscillator discussed e g in a recent paper by gadella glasser and nieto also introduced by p s ba as a perturbation of the one dimensional free laplacian and recently investigated in that context by golovaty hryniv and zolotaryuk we investigate in detail the spectrum of our perturbed harmonic oscillator the spectral structure differs from that of the one dimensional harmonic oscillator perturbed by an attractive interaction centred at the origin the even eigenvalues are not modified at all by the interaction moreover all the odd eigenvalues regarded as functions of exhibit the rather remarkable phenomenon called level crossing after first producing the double degeneracy of all the even eigenvalues for the value b being the beta function

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

on linearly related sequences of derivatives of orthogonal polynomials abstract we discuss an inverse problem in the theory of standard orthogonal polynomials involving two orthogonal polynomial families p n n and q n n whose derivatives of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 n r i n p n i m m x i 0 m s i n q n i k k x for all n 0 1 2 where m and n are fixed nonnegative integer numbers and r i n and s i n are given complex parameters satisfying some natural conditions let u and v be the moment regular functionals associated with p n n and q n n resp assuming 0 m k we prove the existence of four polynomials m m i and n k i of degrees m m i and n k i resp such that d k m m m i u n k i v i 0 1 the k m th derivative as well as the left product of a functional by a polynomial being defined in the usual sense of the theory of distributions if k m then u and v are connected by a rational modification if k m 1 then both u and v are semiclassical linear functionals which are also connected by a rational modification when k m the stieltjes transform associated with u satisfies a non homogeneous linear ordinary differential equation of order k m with polynomial coefficients

characterization and construction of classical orthogonal polynomials using a matrix approach abstract we introduce a new algebraic approach for the study of orthogonal polynomial sequences we use simple properties of infinite matrices to obtain some basic results in the theory of orthogonal polynomial sequences including several characterizations of the classical sequences some of which are equivalent to the ones obtained by bochner maroni and al salam and chihara we also obtain explicit formulas for the coefficients of the recurrence relations of all the classical orthogonal polynomials in terms of four parameters the polynomial coefficients and the eigenvalues that appear in bochner s differential equation are also found explicitly

on some tridiagonal k toeplitz matrices algebraic and analytical aspects applications in this paper we use the analytic theory for 2 and 3 toeplitz matrices to obtain the explicit expressions for the eigenvalues eigenvectors and the spectral measure associated to the corresponding infinite matrices as an application we consider two solvable models related with the so called chain model some numerical experiments are also included

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