weak commutation relations of unbounded operators and applications four possible definitions of the commutation relation s t 11 of two closable unbounded operators s t are compared the weak sense of this commutator is given in terms of the inner product of the hilbert space h where the operators act some consequences on the existence of eigenvectors of two numberlike operators are derived and the partial o algebra generated by s t is studied some applications are also considered

the topology of deformation spaces of kleinian groups let m be a compact hyperbolizable 3 manifold with nonempty incompressible boundary and let ah 1 m denote the space of conjugacy classes of discrete faithful representations of 1 m i nto psl 2 c the components of the interior mp 1 m of ah 1 m as a subset of the appropriate representation variety are enumerated by the spacea m of marked homeomorphism types of oriented compact irreducible 3 manifolds homotopy equivalent to m in this paper we give a topological enumeration of the components of the closure of mp 1 m and hence a conjectural topological enumeration of the components of ah 1 m we do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable kleinian groups we use this enumeration to exhibit manifolds m for which ah 1 m has inflnitely many components in this paper we begin a study of the global topology of deformation spaces of kleinian groups the basic object of study is the space ah 1 m of marked hyperbolic 3 manifolds homotopy equivalent to a flxed compact 3manifold m the interior mp 1 m of ah 1 m is very well understood due to work of ahlfors bers kra marden maskit sullivan and thurston in particular the components ofmp 1 m are enumerated by topological data namely the set a m of marked compact oriented irreducible 3 manifolds homotopy equivalent to m while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3 manifolds thurston s ending lamination conjecture provides a conjectural classiflcation for elements of ah 1 m by data which are partially topologi

complex earthquakes and teichmuller theory it is known that any two points in teichmuller space are joined by an earthquake path in this paper we show any earthquake path r t s extends to a proper holomorphic mapping of a simplyconnected domain d into teichmuller space where r d c these complex earthquakes relate weil petersson geometry projective structures pleated surfaces and quasifuchsian groups using complex earthquakes we prove grafting is a homeomorphism for all 1 dimensional teichmuller spaces and we construct bending coordinates on bers slices and their generalizations in the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold

extensions of some results of p humbert on bezout s identity for classical orthogonal polynomials in this paper the bezout s identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type differential equations relation with the starting family as well as recurrence relations and explicit representations are given for the bezout s pair extensions to classical orthogonal polynomials of a discrete variable and their q analogues are also presented applications of these results for the representation of the second kind functions are given

momentum operators with gauge potentials local quantization of magnetic flux and representation of canonical commutation relations commutation properties of two dimensional momentum operators with gauge potentials are investigated a notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators in terms of the notion a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the schrodinger representation of the canonical commutation relations

orthogonal polynomials associated polynomials and functions of the second kind abstract a survey is given of the interaction between orthogonal polynomials associated polynomials and functions of the second kind with an emphasis on asymptotic results various formulas are presented in a unified way in terms of wronskians of solutions of linear recurrence relations some of these formulas are classical and go back to the previous century but usually they are hard to locate in the literature some new formulas are also given in particular a formula expressing the derivative of an orthogonal polynomial in terms of the orthogonal polynomials and the associated polynomials

dilations of symmetric operators shifted by a unitary group let q be a symmetric operator and ux a one parameter unitary group in a hilbert space such that ux dom q dom q and uxqux q x for dom q it is shown by a compactness argment that q admits a resolution q xf dx with a positive operator valued measure satisfying uxf ux f x this entails the existence of a selfadjoint extension of q to a larger space satisfying the analogus commutation relation with a suitable extended unitary group this result is generalized to n commuting symmetric operators transformed among each other by a representation of an amenable subgroup of the affine group in rn

self bumping of deformation spaces of hyperbolic 3 manifolds let n be a hyperbolic 3 manifold and b a component of the interior of ah 1 n the space of marked hyperbolic 3 manifolds homotopy equivalent to n we will give topological conditions on n sufficient to give b such that for every sufficiently small neighbourhood v of v b is disconnected this implies that b is not a manifold with boundary

the orthogonality property of the lommel polynomials and a twofold infinity of relations between rayleigh s σ sums making use of a remarkable theorem which expresses a relationship between a certain type of infinite continued fractions and systems of orthogonal polynomials it is proven that the known infinite continued fraction development of the ratio of bessel functions jv 1 z jv z gives rise to an orthogonality property of the lommel polynomials rm v 1z m n when v is real and positive the corresponding weight function which appears to be non negative in the interval of definition is obtained by the application of two successive integral transforms it consists of an infinite series of dirac functions whose singularities are distributed symmetrically around the origin on the real axis in such a manner that the origin is their limit point on both sides for any positive v the lommel polynomials form a system of so called orthogonal polynomials of a discrete variable the orthogonality property may also be conveniently expressed by means of a stieltjes integral one of its corollaries is a twofold infinity of linear relations between the sums v r defined by v r n 1 1 jv n2r with v 1 r0 r n0 in which jv n represents the nth positive zero of jv z another by product consists of a complement to a theorem of hurwitz concerning the nature and the position of the zeros of the lommel polynomials written as gm v z in the modified notation of the mentioned author from this study also result two interesting approximations of jv 1 applicable for v 1 1

results on the associated jacobi and gegenbauer polynomials abstract recurrence relations for the coefficients of the jacobi polynomial form of the associated jacobi polynomials p n x c are given for certain values of explicit formulae for these coefficients are obtained in particular the classical watson formula for the first associated gegenbauer polynomials is generalized

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

non commutative moment problems we are concerned with two problems called the generalized moment problem gmp and the quantum moment problem qmp both problems can be considered as non commutative versions of the classical moment problem et g be a connected lie group and 7b e g the universal enveloping algebra of the lie algebra of g suppose f is a given linear functional on 7b e g and is a fixed unitary representation of g we then ask gmp when do there exist a unitary representation u of g and a cyclic c vector for u such that f x x 7b e g qmp when is f of the form f x trdv x x 7b e g where is an operator such that dv x is of trace class for every x 7b e g

factorization of fourth order differential equations for perturbed classical orthogonal polynomials we factorize the fourth order differential equations satisfied by the laguerre hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as the rth associated for genetic r the general co recursive the general co recursive associated the general co dilated and the general co modified classical orthogonal polynomials moreover we find four linearly independent solutions of the fourth order differential equations and show that the factorization obtained for modifications of classical orthogonal polynomials is still valid with some minor changes when the polynomial family modified is semi classical finally we extend the validity of the results obtained for the associated classical orthogonal polynomials with integer order of association from integers to reals

properties of zeros of orthogonal polynomials and related functions abstract there are many results in the literature on orthogonal polynomials concerning the way in which the zeros of a polynomial change as a parameter changes for the classical polynomials there are results due to markoff 1886 and stieltjes 1886 on the monotonic variation of zeros of jacobi and gegenbauer ultraspherical polynomials with respect to parameters our purpose here is to describe some recent results concerning formulas for the derivatives of zeros with respect to a parameter such formulas and their consequences are fairly well developed for bessel functions a limiting case of some of the classical orthogonal polynomials and have recently been explored for some orthogonal polynomials and related functions two general methods will be described the first use extensively by ismail the author and others is based on the hellmann feynman theorem which gives the derivative of an eigenvalue with respect to a parameter it can be applied in either a differential equations or a recurrence relations setting the second method developed in recent work of elbert and the author and applicable to suitable solutions of second order ordinary linear differential equations originates in work of richardson 1918 from this approach we obtain formulas for dc d where c c is a zero of the ultraspherical polynomial p n x these recover the known result that dc d 1 2 we get similar results for certain hermite functions these results are closely related to nicholson type formulas due to durand 1975 for sums of squares of solutions of second order linear ordinary differential equations nicholson type formulas are named after the corresponding formulas for bessel functions due to nicholson

on co recursive associated laguerre polynomials abstract in this note we give results on co recursive associated laguerre polynomials in particular explicit form orthogonality measure and fourth order differential equations satisfied by the polynomials are presented with special attention to some simple limiting cases

minimal solutions of three term recurrence relations and orthogonal polynomials we observe that the well known recurrence relation pn x z z a pn z bnp l z for orthogonal polynomials admits a minimal solution if z is outside the spectrum of the mass distribution ds t with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined the minimal solution indeed is f z m ds t z t and can be computed accurately by means of the author s continued fraction algorithm an application is made to special gauss type quadrature formulas 1 introduction minimal solutions of three term recurrence relations and their computational implications are discussed systematically in 7 effective algorithms for computing minimal solutions have been developed in 7 16 17 and continue to be the subject of further study see e g 14 26 5 2 24 25 3 with these powerful algorithms at hand it seems desirable to delineate large classes of recurrence relations for which the presence of minimal solutions can be ascertained and the minimal solution itself identified even more desirable is an understanding of the deeper reasons for the occurrence of minimal solutions in terms of intrinsic features of the subject area in which they arise few attempts have been made along these lines the work of thacher 22 23 on power series solution of linear differential equations with polynomial coeffi cients may be considered a beginning inasmuch as the presence of minimal solutions of the recurrence relation satisfied by the expansion coefficients is conjectured to be related in a specified way to the type of singularities in the differential equation here we wish to consider the recurrence relation

finite perturbations of orthogonal polynomials in this paper we study orthogonal polynomials pn which arise from a given system of orthogonal polynomials pn by a finite perturbation of the recurrence coefficients i e the recurrence coefficients n n 1 of pn respectively n n 1 of pn are related to each other by n l n m and n 1 l n 1 m for n n where l and m are fixed nonnegative integers closed expressions for the polynomial pn in terms of pn and its associated polynomial pn 1 1 as well as for the orthogonality measure of pn in terms of the orthogonality measure of pn are given in fact the results are presented for the most general case when is an arbitrary not necessarily positive measure

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

non realizability and ending laminations proof of the density conjecture we give a complete proof of the bers sullivan thurston density conjecture in the light of the ending lamination theorem it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds the ending invariants are either riemann surfaces or filling laminations supporting masur domain measured laminations and satisfying some mild additional conditions with any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence we derive the necessary compactness theorem combining the rips machine with non existence results for certain small actions on real trees of free products of surface groups and free groups we prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface in order to be able to apply the ending lamination theorem we have to prove that this algebraic limit has the desired topological type and that these non realized laminations are ending laminations that this is the case is the main novel technical result of this paper loosely speaking we show that a filling masur domain lamination which is not realized is an ending lamination

laminations mesurees de plissage des varietes hyperboliques de dimension 3 for a hyperbolic metric on a 3 dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics the locus and intensity of this bending is described by a measured geodesic lamination which is a topological object we consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination

a discrete approach to monotonicity of zeros of orthogonal polynomials we study the monotonicity with respect to a parameter of zeros of orthogonal polynomials our method uses the tridiagonal jacobi matrices arising from the three term recurrence relation for the polynomials we obtain new results on monotonicity of zeros of associated laguerre al salam carlitz meixner and pollaczek polynomials we also derive inequalities for the zeros of the al salam carlitz and meixner polynomials

interlacing properties of zeros of associated polynomials abstract let w be a nonnegative integrable weight function on 1 1 and let pn 1 x xn 1 be the polynomial of degree n 1 orthogonal with respect to w furthermore let pn 1 x xn denote the polynomials associated with pn 1 and pn 1 x2 x xn the polynomials orthogonal with respect to the weight function 1 x2 w x in this paper we give necessary and sufficient conditions such that the zeros of pn 1 and pn 1 x2 strictly interlace on 1 1 for large n in particular this problem is studied for the jacobi weights w x 1 x 1 x 1 in this case p n 1 x 2 p n 1 n 1 for a large class of parameters including e g the ultraspherical case it is shown that the interlacing property holds for each n n also a fairly complete description of the parameters for which the interlacing property does not hold is given

theory of recursive generation of systems of orthogonal polynomials an illustrative example abstract pending its publication in full detail the theory of recursive generation of systems of orthogonal polynomials is verbally described and illustrated by means of a typical example of practical application for this purpose the theory is applied to the polynomials of legendre chosen as initially given orthogonal sequence

constrained d and d_1 optimal designs for polynomial regression in the common polynomial regression model of degree m we consider the problem of determining the d and d 1 optimal designs subject to certain constraints for the d 1 efficiencies in the models of degree m j m j 1 m k m j 0 k 0 given we present a complete solution of these problems which on the one hand allow a fast computation of the constrained optimal designs and on the other hand give an answer to the question of the existence of a design satisfying all constraints our approach is based on a combination of general equivalence theory with the theory of canonical moments in the case of equal bounds for the d 1 efficiencies the constrained optimal designs can be found explicitly by an application of recent results for associated orthogonal polynomials

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

canonical quantization on a doubly connected space and the aharonov bohm phase abstract we consider the canonical quantization schrodinger representation on a doubly connected space r r 2 x y x2 y2 r2 r 0 we show that when we employ 2 dimensional orthogonal coordinates ox1x2 there are uncountably many different self adjoint extensions puj of pj i xj j 1 2 and none of the pairs pj qj j j 1 2 qj xj satisfies the weyl relation then we construct a new canonical pair of canonical momentum and position operators so that the pair can satisfy the weyl relation by using the streamline coordinates as its application in the weyl relation with respect to the pair of the mv momentum and position operators by the above new canonical pair we find the aharonov bohm phase

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

multiple bumping of components of deformation spaces of hyperbolic 3 manifolds let m be a hyperbolizable 3 manifold with nonempty incompressible boundary of negative euler characteristic suppose that b 1 b k is a collection of components of the interior of the space of complete marked hyperbolic 3 manifolds homotopy equivalent to m such that for any i j inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we prove that there is a geometrically finite hyperbolic structure on int m which is in the closure of each b i moreover we show that this structure can be constructed so as to admit quasiconformal deformations which also lie in the closure of every b i

on continuous extensions of grafting maps the definition of the grafting operation for quasifuchsian groups is extended by bromberg preprint to all b groups in this paper we show that the extended grafting maps behave as continuous maps for every sequence which converges standardly to a boundary group although the maps are not continuous in general as a consequence of this result we extend goldman s grafting theorem for quasifuchsian groups to all boundary b groups

stieltjes sums for zeros of orthogonal polynomials stieltjes considered sums of reciprocals of differences of zeros of a solution of a homogeneous second order linear differential equation here we re examine the derivation of these sums with a view to extending the class of differential equations to which the theory applies and including sums involving the zeros of the derivatives as well as those of the polynomials themselves

zeros of jacobi functions of second kind the number of zeros in 1 1 of the jacobi function of second kind qn x 1 i e the second solution of the differential equation 1 x2 y x 2 x y x n n 1 y x 0 is determined for every n n and for all values of the parameters 1 and 1 it turns out that this number depends essentially on and as well as on the specific normalization of the function qn x interlacing properties of the zeros are also obtained as a consequence of the main result we determine the number of zeros of laguerre s and hermite s functions of second kind

dual birth and death processes and orthogonal polynomials we discuss connections between birth and death processes and orthogonal polynomials we survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes birth and death processes with linear birth and death rates are studied in some detail we also mention some results concerning birth and death processes whose transition rates are rational functions

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

robust designs for polynomial regression by maximizing a minimum of d and d1 efficiencies in the common polynomial regression of degree m we determine the design which maximizes the minimum of the d efficiency in the model of degree m and the d 1 efficiencies in the models of degree m j m k j k 0 given the resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness of fit of the assumed model of degree m by testing the highest coefficients in the polynomials of degree m j m k our approach is based on a combination of the theory of canonical moments and general equivalence theory for minimax optimality criteria the optimal designs can be explicitly characterized by evaluating certain associated orthogonal polynomials

fourth order differential equation satisfied by the associated of any order of all classical orthogonal polynomials a study of their distribution of zeros abstract the first associated numerator polynomials of all classical orthogonal polynomials satisfy one fourth order differential equation valid for the four classical families but for the associated of arbitrary order the differential equations are only known separately in this work we introduce a program built in mathematica symbolic language which is able to construct the unique differential equation satisfied by the associated of any order of the classical class then we use this differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of the kth power of zeros which are closely related with the moments of such a distribution

umbral calculus discretization and quantum mechanics on a lattice umbral calculus deals with representations of the canonical commutation relations we present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of lie algebras on a lattice related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus this observation then suggests various generalizations of these examples a special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail

algebraic limits of kleinian groups which rearrange the pages of a book in this paper we give examples of two new phenomena in kleinian groups we rst exhibit a sequence of homeomorphic marked hyperbolic 3 manifolds whose algebraic limit is not homeomorphic to any element in the sequence we then use this construction to exhibit situations where the space of convex co compact representations of a given 3 manifold group has many components but its closure is connected let m be a compact irreducible oriented 3 manifold and let d 1 m denote the space of all discrete faithful representations of 1 m into psl2 c a sequence of representations n d 1 m converging to d 1 m gives rise to a sequence n n h n 1 m of hyperbolic 3 manifolds each of which is homotopy equivalent to m the sequence n n is said to converge algebraically to n h 1 m see 7 13 14 for more information about algebraic convergence of kleinian groups in many situations see 1 6 15 24 25 27 it has been shown that n n must be homeomorphic to n for all large enough n and we had suspected that this would always be the case in this paper we give a collection of examples where n n is not homeomorphic to n for any n our sequences are quite well behaved the n 1 m are convex co compact and mutually quasiconformally conjugate and the algebraic limit 1 m is geometrically nite in our examples m is obtained by gluing a collection of i bundles to a solid torus along a family of parallel annuli these manifolds are particularly simple examples of books of i bundles see 9 where to explain the terminology one should think of the solid torus as the binding and the i bundles

quasiconformal homeomorphisms and dynamics ii structural stability implies hyperbolicity for kleinian groups sous groupe de psl 2 c familles holomorphes d homeomorphismes de c lemme champs de ligne invariants familles holomorphes de groupes abstraitement isomorphes car finiment genere la stabilite structurelle implique geometriquement fini ou rigide theoreme a avec paraboles stabilite structurelle au sens de la dynamique differentiable

exotic projective structures and quasi fuchsian space ii let p s be the space of projective structures on a closed surface s of genus g 1 and let q s be the subset of p s of projective structures with quasi fuchsian holonomy it is known that q s consists of infinitely many connected components in this article we show that the closure of any exotic component of q s is not a topological manifold with boundary and that any two components of q s have intersecting closures

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

the visual core of a hyperbolic 3 manifold in this note we introduce the notion of the visual core of a hyperbolic 3 manifold n bf h 3 gamma and explore some of its basic properties we investigate circumstances under which the visual core cal v n of a cover n bf h 3 gamma of n embeds in n via the usual covering map pi n rightarrow n we go on to show that if the algebraic limit of a sequence of isomorphic kleinian groups is a generalized web group then the visual core of the algebraic limit manifold embeds in the geometric limit manifold finally we discuss the relationship between the visual core and klein maskit combination along component subgroups

on pairs of difference operators satisfying d x id dedicated to the memory of gerda dorfmeister different finite difference replacements for the derivative are analyzed in the context of the heisenberg commutation relation the type of the finite difference operator is shown to be tied to whether one can naturally consider d and x to be self adjoint and skew self adjoint or whether they have to be viewed as creation and annihilation operators the first class generalizing the central difference scheme is shown to give unitary equivalent representations for the second case we construct a large class of examples generalizing previously known difference operator realizations of d x id

fourth order differential equation for the co modified of semi classical orthogonal polynomials abstract we give a new derivation of the fourth order differential equation satisfied by the co modified orthogonal polynomials of any semi classical family of orthogonal polynomials this procedure generalizes a technique already used in the co recursive and co dilated cases for the classical cases we give explicitly the corresponding differential equation in a determinantal form

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

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

the grafting map of teichmuller space one of the underlying principles in the study of kleinian groups is that aspects of the complex projective geometry of quotients of by these groups reflect properties of the three dimensional hyperbolic geometry of the quotients ofh by these groups yet even though it has been over thirty five years since lipman bers exhibited a holomorphic embedding of the teichmuller space of riemann surfaces in terms of the projective geometry of a teichmuller space of quasi fuchsian manifolds no corresponding embedding in terms of the three dimensional hyperbolic geometry has been presented one of the goals of this paper is to give such an embedding this embedding is straightforward and has been expected for some time ta97 mc98 to each member of a bers slice of the space qf of quasi fuchsian 3manifolds we associate the bending measured lamination of the convex hull facing the fixed conformal end the geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting an operation on projective structures on surfaces that traces its roots back at least to klein kl33 p 230 50 with a modern history developed by many authors ma69 he75 fa83 st83 go87 gkm00 ta97 mc98 the main technical tool in our proof that bending measures give coordinates for bers slices and the second major goal of this paper is the completion of the proof of the grafting conjecture this conjecture states that for a fixed measured lamination the self map of teichmuller space induced by grafting a surface along is a homeomorphism of teichmuller space our contribution to this argument is a proof of the injectivity of the grafting map while the principal application of this result that we give is to geometric coordinates on the bers slice of qf one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of kleinian groups see 5 2 thus we feel that this result is of interest in its own right we now state our results and methods more precisely throughout s will denote a fixed differentiable surface which is closed orientable and of genus g 2 let tg be the corresponding teichmuller space of marked conformal structures on s and let pg denote the deformation space of complex projective structures on s see 2 for definitions

fourth order differential equations satisfied by the generalized co recursive of all classical orthogonal polynomials a study of their distribution of zeros the unique fourth order differential equation satisfied by the generalized co recursive of all classical orthogonal polynomials is given for any but fixed level of recursivity up to now these differential equations were known only for each classical family separately and also for a specific recursivity level moreover we use this unique fourth order differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of powers of their zeros which are closely related with the moments of such distribution both results are obtained with the help of two programs built in mathematica symbolic language

integrals of jacobi functions in this paper we study f 1 x a 1 x bp a x dx and its partial derivatives with respect to a and b where p a x is the jacobi function our expressions generalize the work of blue gautschi and gatteschi the results are useful to derive integration formulas for integrands with algebraic and logarithmic singularities

a generalized weyl relation approach to the time operator and its connection to the survival probability the time operator an operator which satisfies the canonical commutation relation with the hamiltonian is investigated on the basis of a certain algebraic relation for a pair of operators t and h where t is symmetric and h self adjoint this relation is equivalent to the weyl relation in the case of self adjoint t and is satisfied by the aharonov bohm time operator t0 and the free hamiltonian h0 for the one dimensional free particle system in order to see the qualitative properties of t0 the operators t and h satisfying this algebraic relation are examined in particular it is shown that the standard deviation of t is directly connected to the survival probability and h is absolutely continuous hence it is concluded that the existence of the operator t implies the existence of scattering states it is also shown that the minimum uncertainty states do not exist other examples of these operators t and h than the one dimensional free particle system are demonstrated

random walk with long range interaction with a barrier and its dual exact results we consider the random walk on z 0 1 with up and down transition probabilities given the chain is in state x 1 2 1 p x 12 1 d2x d andq x 12 1 d2x d here d 1 is a real tuning parameter we assume that this random walk is reflected at the origin for d 0 the walker is attracted to the origin the strength of the attraction goes like d2x for large x and so is long ranged for d 0 the walker is repelled from the origin this chain is irreducible and periodic it is always recurrent either positive or null recurrent using karlin mcgregor s spectral representations in terms of orthogonal polynomials and first associated orthogonal polynomials exact expressions are obtained for first return time probabilities to the origin excursion length eventual return contact probability excursion height and spatial moments of the walker all exhibit power law decay in some range of the parameter d in the study an important role is played by the wall duality relation for birth and death chains with reflecting barrier some qualitative aspects of the dual random walk obtained by interchanging p x and q x are therefore also included

classification of certain pairs of operators p q satisfying p q i id abstract in a separable hilbert space a certain class of pairs of operators p q satisfying the born heisenberg commutation relation p q i id on a dense domain is investigated this class is essentially defined by requiring q bounded and self adjoint p symmetric and p q we show that q is absolutely continuous and that p can be thought of as a first order differential operator the class considered contains the pair angle and angular momentum l z it is expected that the methods of this paper can be applied to more general classes of operators p q including the schrodinger case

quantum mechanics as noncommutative symplectic geometry the authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation this leads to a mathematical framework in which quantum mechanics can be understood as noncommutative symplectic geometry they point out the possibility to describe interactions geometrically as deformations of the differential calculus

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

properties of the polynomials associated with the jacobi polynomials abstract power forms and jacobi polynomial forms are found for the polynomials w associated with jacobi polynomials also some differential difference equations and evalua tions of certain integrals involving wii 1 are given 1 introduction let w be a nonnegative function defined on an interval a b forwhich all momentsm j x w x dx n 0 1 exist and are finite m0 0 let be the monic polynomials orthogonal on a b with respect to w the polynomials 1 1 q x n pax rztt wu at pn t jt n 0 1 j a x are called the polynomials associated with the pn as is well known pn x a x are linearly independent solutions of the recurrence formula i 2 yn i x a y bny x q 0 1 where l3 an xpn p p pn n 0 o o b pn pn pn x pn x n l and f g f f x g x w x dx ja the initial conditions arep i x 0 pa x 1

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

the weight functions generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the jacobi and the gegenbauer polynomials abstract in the introduction the main object of the paper namely the calculation and study of the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the jacoby and the gegenbauer polynomials is briefly discussed section 2 contains the underlying theory restricted to the citation of a few general formulas regarding the orthogonality property of associated polynomials of the second kind section 3 which deals with the application to the case of the jacobi polynomials p n z n n starts with the construction of the orthogonality relation and the weight function 1 of the associated polynomials for arbitrary gt 1 the denominator of the fraction representing 1 comprises the principal value of an integral which is studied in detail in order to bring it in an explicit form most suited for practical use attention is also paid to the associated functions of the second kind and some of their properties next in a subsection 1 is expressed solely in terms of elementary functions in a number of special cases involving subsets of pairs when 1 1 n gt 1 1 1 gt0 and 1 0 gt 1 in another subsection the generating function for the associated polynomials is calculated and the analysis of the result comprises some related formulas in the final subsection the moments of the weight function 1 with respect to 0 1 and 1 are expressed in terms of the corresponding moments of the weight function of the jacobi polynomials both in finite determinantal form and by means of a recurrence relation in section 4 which concerns the application to the case of the gegenbauer polynomials c v n z n n the advantages of a new definition of the associated polynomials of the second kind are at first expounded then follow the orthogonality relation and the weight function 1 for arbitrary v gt 1 2 next 1 is calculated in terms of elementary functions for all integer and half odd integer values of v exceeding 1 2 in the former case a remarkable simplification occurs in the denominator of the fraction representing 1 after that follow three subsections devoted respectively to the functions of the second kind associated with the gegenbauer polynomials two different generating functions for the associated polynomials and related formulas and the moments of the weight function 1 the paper ends with two appendices in which certain types of sums are reduced to a one term expression

legendre polynomials of the second kind fourier series and lagrange interpolation abstract the distribution of the zeros of jacobi polynomials of the second kind p m w is studied and lower and upper bounds for the corresponding christoffel functions are given fourier expansions in the system p m w are also considered moreover some theorems on the convergence of lagrange interpolating polynomials based on the zeros of p m w are proved

recurrence relations for hypergeometric functions of unit argument we show that the generalized hypergeometric function p 1 p 3fp 2 nn 2 1 a n 0 satisfies a nonhomogeneous recurrence relation of order p a where a 0 when p 3 2 1 is balanced and a 1 otherwise also for us dq n a 4 2 q l n dq 2n a 1 1 n o a homogeneous recurrence relation of order q 1 is given 1 notation the generalized hypergeometric series a a l a 1 00 p 1 p 1 1 p t ail b p e al ai kz k k h b1 k where 1 2 a k f a k f a converges for iz i 0 let x ai i 1 2 p b1 j 1 2 p 2 ck k 1 2 q 2 di i 1 2 q be complex constants we define the functions g 1 n t n 1 m m 2n x 2m 2m n a m m 2n x m t m n bp 21 x m n r 1 bp 2 3 1 f m 2n x m t n m bp 2 p 4p 3l 2n x 2m 1 n m 1 bp 2 f 2n a m t n m ap 1 m 2n x t 1 n m 1 a 2n x 2m 1 n m a j h 3 n t 2n x m n b 1 m n x m s 1 m n 2n a n 1 cq t m q 4 q 3 2n a q 2 n cq 2 3 2 2n x m n dq rn m n cq 2 1 m l2n m 1 n 1 dq xq 2 q l 2n x q 2 n dq here b 1 ci for i q 2 we assume the parameters to be such that all expressions make sense by applying a lemma of wimp 8 we get the alternative forms for gm hm gm n t l p n 1 m n 2n x 2m 2n x t i t n x m m 2n x m t m n 1 bp 2 x t r n x t 1 2 r t 2n x m t n x t 2 bp 2 p 4p3 2n x t 1 n x t 1 bp 2 2n x m t n x t 1 ap f t rn mr t 1 2n x m t 1 n x t 2 a xp 2fp l2 2n x t 1 n x t 1 ap j this content downloaded from 157 55 39 116 on sun 18 sep 2016 06 20 49 utc all use subject to http about jstor org terms 524 stanislaw lewanowicz l1 q 2n x q 2 n 1 m n x m 2n x m mi n cq 2

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

parameter based fisher s information of orthogonal polynomials the fisher information of the classical orthogonal polynomials with respect to a parameter is introduced its interest justified and its explicit expression for the jacobi laguerre gegenbauer and grosjean polynomials found

completely positive conjugate bilinear maps on partial algebras the set of invariant completely positive conjugate bilinear maps on an arbitrary unital partial algebra is introduced and characterized by considering a natural order structure on this set all order intervals in it having the zero map as minorant are characterized and a certain extremal problem is solved in particular a radon nikodym theorem for members of the set is deduced finally an extension theorem for completely positive but not necessarily invariant conjugate bilinear maps on unital partial algebras is obtained

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

convergence and divergence of kleinian punctured torus groups in this paper we give a necessary and sufficient condition in which a sequence of kleinian punctured torus groups converges this result tells us that every exotically convergent sequence of kleinian punctured torus groups is obtained by the method due to anderson and canary invent math 1996 thus we obtain a complete description of the set of points at which the space of kleinian punctured torus groups self bumps we also discuss hausdorff limits of sequences of bers slices

associated jacobi laurent polynomials abstract the jacobi laurent polynomials have been introduced by hendriksen and van rossum 1986 in the present paper explicit formulas for the orthogonal laurent polynomials satisfying the recurrency for the jacob laurent polynomials with n replaced by n b are given these new orthogonal laurent polynomials are called associated jacobi laurent polynomials using these associated laurent polynomials the denominator and the numerator of certain two point pade approximants to the pair of functions z f a b 1 c b 1 z f a b c b z at o and c b a b 1 f c 1 b 1 a b 2 z 1 f c 1 b 1 z l at are given also some confluent cases are considered

the normal distribution is infinitely divisible abstract we prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution we study the voiculescu transform first by giving a survey of its combinatorial implications and then analytically including a proof of free infinite divisibility in fact we prove that a sub family of askey wimp kerov distributions are freely infinitely divisible of which the normal distribution is a special case at the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution the others being the cauchy distribution and the free 1 2 stable distribution

cores of hyperbolic 3 manifolds and limits of kleinian groups troels jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic kleinian groups agree if there are no new parabolics in the algebraic limit we prove that this conjecture holds in most cases in particular we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non empty we further show with the same assumptions that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit as a corollary we verify the conjecture for finitely generated kleinian groups which are not non trivial free products of surface groups and infinite cyclic groups these results are extensions of similar results for purely loxodromic groups thurston previously established these results in the case when the kleinian groups are freely indecomposable using different techniques from ours ohshika has proven versions of these results for purely loxodromic function groups

octobasic laguerre polynomials and permutation statistics abstract a set of orthogonal polynomials with eight independent q s is defined which generalizes the laguerre polynomials the moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics specializing these statistics gives many other well known sets of combinatorial objects and relevant statistics the specializations are studied with applications to classical orthogonal polynomials and equidistribution theorems for statistics

quick construction of recurrence relations for the jacobi coefficients abstract recurrence relations are given for the jacobi series coefficients of functions which satisfy linear differential equations of first to third order with polynomial coefficients

examples of relative deformation spaces that are not locally connected we provide an infinite family of pared manifolds with the property that the relative deformation spaces of hyperbolic structures on these manifolds are not locally connected this is a natural extension of the recent result of bromberg that shows the space of kleinian punctured torus groups is not locally connected