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

linear perturbations of differential or difference operators with polynomials as eigenfunctions abstract this paper deals with one parameter linear perturbations of a family of polynomials pn x n 0 with deg pn x n of the form p n x p n x q n x where is a real parameter and qn x n 0 are polynomials with deg qn x n let the polynomials pn x n 0 be eigenfunctions of a linear differential or difference operator l a with eigenvalues n n 0 the purpose of this p to derive necessary and sufficient conditions for the polynomials qn x n 0 such that the polynomials pn x n 0 are eigenfunctions of a linear difference or differential operator possibly of infinite order of the form l a with eigenvalues n n n 0

differential operators having sobolev type laguerre polynomials as eigenfunctions new developments abstract in this paper we consider polynomials orthogonal with respect to an inner product which consists of the classical laguerre inner product combined with two linear perturbations of sobolev type at x 0 we derive linear differential operators of a specific form and usually of infinite order having these polynomials as eigenfunctions in the case is a nonnegative integer one of the operators is of finite order

differential and difference operators having orthogonal polynomials with two linear perturbations as eigenfunctions abstract in this paper we consider the polynomials pn x n 0 orthogonal with respect to a certain symmetric bilinear form of sobolev type these polynomials are the result of two linear perturbations to the orthogonal polynomials pn x n 0 eigenfunctions of a linear differential or difference operator l we show that the polynomials pn x n 0 are eigenfunctions of one or more linear differential or difference operators possibly of infinite order of the form l a b c

a note on the koekoek s differential equation for generalized jacobi polynomials abstract in a recent paper differential equations for generalized jacobi polynomials submitted for publication koekoek and koekoek discovered a linear differential equation for the polynomials pn m n x n 0 which are orthogonal on 1 1 with respect to 0 1 2 2 1 1 1 1 x 1 x m x 1 n x 1 1 m n 0 this differential equation is of infinite order except in a number of cases it is the purpose of this note to reprove and interpret the results of the koekoeks in the finite order cases in a short and easy way

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

the q version of a theorem of bochner askey and wilson 1985 found a family of orthogonal polynomials in the variable s k 1 2 k 1 k that satisfy a q difference equation of the form a k p n s qk p n s k b k p n s k q p n s k theta n p n s k n 0 1 we show here that this property characterizes the askey wilson polynomials the proof is based on an operator identity of independent interest this identity can be adapted to prove other characterization results indeed it was used in grunbaum and haine 1996 to give a new derivation of the result of bochner alluded to in the title of this paper we give the appropriate identity for the case of difference equations leading to the wilson polynomials but pursue the consequences only in the case of q difference equations leading to the askey wilson and big q jacobi polynomials this approach also works in the discrete case and should yield the results in leonard 1982

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

orthogonal polynomial eigenfunctions of second order partial differerential equations in this paper we show that for several second order partial differential equations l u a x y uxx 2b x y uxy c x y uyy d x y ux e x y uy atu which have orthogonal polynomial eigenfunctions these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable this is important since otherwise it is very difficult to explicitly find formulas for these polynomial solutions from this observation and characterization we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by h l krall and sheffer in their seminal work in 1967 moreover from our approach we can answer some open questions raised by krall and sheffer

the search for differential equations for certain sets of orthogonal polynomials abstract we look for differential equations of the form i 0ci x y i x 0 where the coefficients ci x i 0 are continuous functions on the real line and where ci x i 1 are independent of n for the generalized jacobi polynomials p m nn x n 0 and for generalized laguerre polynomials l m nn x n 0 which are orthogonal with respect to an inner product of sobolev type we use a method involving computer algebra packages like maple and mathematica and we will give some preliminary results

differential operators having sobolev type jacobi polynomials as eigenfunctions in a recent paper koekoek and koekoek j comput appl math 126 2000 1 31 discovered a linear differential equation for the jacobi type polynomials pn m n x n 0 which are orthogonal on 1 1 with respect to 2 2 1 1 1 1 x 1 x m x 1 n x 1 1 m n 0 0 1 if m2 n2 0 this differential equation is of finite order in the following cases 1 m 0 n 0 and 0 1 2 2 m 0 n 0 and 0 1 2 3 m 0 n 0 and 0 1 2 in this paper the result will be generalized to sobolev type jacobi polynomials

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

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

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

some functions that generalize the krall laguerre polynomials let l alpha be the semi infinite tridiagonal matrix associated with the three term recursion relation satisfied by the laguerre polynomials with weight function 1 gamma alpha 1 z alpha e z alpha 1 on the interval 0 infinity we show that when alpha is a positive integer by performing at most alpha successive darboux transformations from l alpha we obtain orthogonal polynomials on 0 infinity with weight distribution 1 gamma alpha k 1 z alpha k e z sigma j 1 k s j delta k j z with 1 less than or equal to k less than or equal to alpha we prove that as a consequence of the rational character of the darboux factorization these polynomials are eigenfunctions of a finite order differential operator our construction calls for a natural bi infinite extension of these results with polynomials replaced by functions of which the semi infinite case is a limiting situation c 1999 elsevier science b v all rights reserved

differential operators having symmetric orthogonal polynomials as eigenfunctions abstract let the polynomials pn x n 0 orthogonal with respect to a symmetric positive definite moment functional be eigenfunctions of a linear differential operator l we consider the orthogonal polynomials pn x n 0 and pn v x n 0 which are obtained by adding one resp two symmetric sobolev type terms to in all the cases we derive a representation for the polynomials and show that they are eigenfunctions of one or more linear differential operators mostly of infinite order of the form l a resp l a v b v c further it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

some connection and linearization problems for polynomials in and beyond the askey scheme the connection problem is considered in a hypergeometric function framework for i the two most general families of polynomials belonging to the askey scheme wilson and racah and ii some generalized laguerre and jacobi polynomials falling outside that scheme sister celine cohen and prabhakar jain which are relevant to the study of quantum mechanical systems and include as particular cases the generalizations of the classical families with sobolev type orthogonality in addition using the same method three new linearization like formulae for the gegenbauer polynomials are also derived a linearization formula that generalizes the m n case of dougall s formula the analogue of the m n case of nielsen s inverse linearization formula for hermite polynomials and a connection formula for the squares closed analytical formulae for the corresponding connection and linearization coe cients are given in terms of hypergeometric functions of unit argument which at times can be further simpli ed and expressed as single hypergeometric terms c 2001 published by elsevier science b v

a general left definite theory for certain self adjoint operators with applications to differential equations we show that any self adjoint operator a bounded or unbounded in a hilbert space h v that is bounded below generates a continuum of hilbert spaces hr r 0 and a continuum of self adjoint operators ar r 0 for reasons originating in the theory of differential operators we call each hr the rth left definite space and each ar the rth left definite operator associated with h a each space hr can be seen as the closure of the domain d ar of the self adjoint operator ar in the topology generated from the inner product arx y x y d ar furthermore each ar is a unique self adjoint restriction of a in hr we show that the spectrum of each ar agrees with the spectrum of a and the domain of each ar is characterized in terms of another left definite space the hilbert space spectral theorem plays a fundamental role in these constructions we apply these results to two examples including the classical laguerre differential expression l in which we explicitly find the left definite spaces and left definite operators associated with a the self adjoint operator generated by l in l2 0 t e t having the laguerre polynomials as eigenfunctions

variations on a theme of heine and stieltjes an electrostatic interpretation of the zeros of certain polynomials we show a way to adapt the ideas of stieltjes to obtain an electrostatic interpretation of the zeroes of a large class of orthogonal polynomials

differential equations and sobolev orthogonality abstract consider sobolev orthogonal polynomials which are orthogonal relative to a sobolev bilinear form r p x 1 x d x r p x q x d x where d x and d v x are signed borel measures with finite moments we give necessary and sufficient conditions under which such orthogonal polynomials satisfy a linear spectral differential equation with polynomial coefficients we then find a sufficient condition under which such a differential equation is symmetrizable these results can be applied to sobolev laguerre polynomials found by koekoek and meijer

electrostatic interpretation for the zeros of certain polynomials and the darboux process abstract the purpose of this paper is to extend the electrostatic interpretation of stieltjes to the zeros of a larger class of important polynomials to keep things simple i consider polynomials that are obtained from those considered by stieltjes by means of the so called darboux process applied to the difference operator featuring in their three term recursion relation it will turn out that some of the beauty in the results of stieltjes is preserved in this process it would be interesting to interpret the darboux process at the electrostatic level and this paper gives experimental data that might help in this direction

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

unitary transformation approach to the cyclic evolution of su 1 1 and su 2 time dependent systems and geometrical phases using the time dependent unitary transformation instead of the invariant operator the solutions of su 1 1 and su 2 time dependent quantum systems are obtained it is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the hamiltonian and the other correspond to the floquet operator which gives the cyclic states and their associated phases over one period of the evolution the non adiabatic berry s or aharonov anandan phases are determined totally by such a unitary transformation

asymptotic properties of sobolev orthogonal polynomials abstract in this report we will survey some of the main ideas and tools which appeared recently in the study of the analytic properties of polynomials orthogonal with respect to inner products involving derivatives although some results on weak asymptotics are mentioned the strong outer asymptotics constitutes the core of the paper both the discrete and the continuous cases are considered and several open problems and conjectures are posed

sobolev orthogonal polynomials and spectral differential equations we find necessary and sufficient conditions for a spectral differential equation n ln y x 52ti x yw x kny x 1 to have sobolev orthogonal polynomials of solutions which are orthogonal rel ative to the sobolev pseudo inner product c t p q pqdn i p q du

compact derivations relative to semifinite von neumann algebras soit a b h une algebre de von neumann soit une derivation de a dans b h on considere l ideal norme clos des 2 cotes j b engendre par les projections finies d une algebre de von neumann b on etudie la sous algebre initiale a de b qui est une derivation de a dans j b implantee par un element de j b

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result

jacobi sobolev type orthogonal polynomials second order differential equation and zeros abstract we obtain an explicit expression for the sobolev type orthogonal polynomials qn associated with the inner product p q 1 1 p x q x p x dx a 1 p 1 q 1 b 1 p 1 q 1 a 2 p 1 q 1 b 2 p 1 q 1 where p x 1 x 1 x is the jacobi weight function 1 a1 b1 a2 b2 0 and p q p the linear space of polynomials with real coefficients the hypergeometric representation 6f5 and the second order linear differential equation that such polynomials satisfy are also obtained the asymptotic behaviour of such polynomials in 1 1 is studied furthermore we obtain some estimates for the largest zero of qn x such a zero is located outside the interval 1 1 we deduce his dependence of the masses finally the wkb analysis for the distribution of zeros is presented

sobolev orthogonality for the gegenbauer polynomials c n n 1 2 n 0 abstract in this work we obtain the property of sobolev orthogonality for the gegenbauer polynomials c n n 1 2 n 0 with n 1 a given nonnegative integer that is we show that they are orthogonal with respect to some inner product involving derivatives the sobolev orthogonality can be used to recover properties of these gegenbauer polynomials for instance we can obtain linear relations with the classical gegenbauer polynomials