the extremal zeros of a perturbed orthogonal polynomials system the purpose of this paper is to study how the extremal zeros of a family of orthogonal polynomials evolve when we perturb the coefficient of the recurrence relation defining the family to this end we shall compare the extremal zeros with the corresponding zeros in the perturbed case

on the support of the measure of orthogonality of a class of orthogonal polynomials for orthogonal polynomials of the formp nq n 1 x q nq n 1 x r nq n x xq n x q 0 x 0 q 1 x 1 where p n 0 q n 0 r email protected r and lim n p n lim n q n 1 2 a general sufficient condition is found such that the support of the measure of orthogonality is the entire interval 1 1 starting from this result more general cases of orthogonal polynomials are studied as a perturbation problem the results are applied to pollaczek polynomials random walk polynomials rwp neutron transport polynomials and generalized co recursive polynomials

perturbation of the coefficients in the recurrence relation of a class of polynomials let p n x n 0 be a system of polynomials satisfying the recurrence relation p 1 x 0 p 0 x 1 p n 1 x h n p n 1 x c n p n x xp n x where h n c n are real sequences and h n 0 n 0 1 2 the co recursive polynomials p n x n 0 satisfy the same recurrence relation except for n 1 where p 1 x x c 0 0 it is well known that the problem of d the zeros of p n x is equivalent to the problem of determining the eigenvalues of a generalized eigenvalue problem tf af where t and a are symmetric matrices in this paper the problem of determining the zeros of the co recursive polynomials is reduced to a perturbation problem of the operators t and a perturbed by perturbations of rank one a function 1 2 k is found k 1 2 n whose zeros are the zeros of p n x k are the zeros of the polynomial p n x of degree n for 0 this function unifies many results concerning interlacing between the zeros of p n x and p n x for 0 moreover we obtain from this function similar results in the unstudied case 0

computation of newton sum rules for associated and co recursive classical orthogonal polynomials abstract by using some preceding results of buendia et al in alfaro et al eds orthogonal polynomials and their applications lecture notes in mathematics vol 1329 springer berlin 1986 pp 222 235 ricci j math phys 34 1993 4884 4891 natalini calcolo 31 1994 127 144 and differential equations of associated see belmedhi ronveaux rend mat 11 1991 313 326 zarzo et al j comput appl math 49 1993 349 359 and co recursive see ronveaux marcellan j comput appl math 25 1989 105 109 ronveaux et al j comput appl 59 1995 295 328 orthogonal polynomials we obtain numerical results for newton sum rules of associated and co recursive laguerre hermite and jacobi polynomials

approximate standard errors in semiparametric models we consider semiparametric models with p regressor terms and q smooth terms we obtain an explicit expression for the estimate of the regression coefficients given by the back fitting algorithm the calculation of the standard errors of these estimates based on this expression is a considerable computational exercise we present an alternative approximate method of calculation that is less demanding with smoothing splines the method is exact while with loess it gives good estimates of standard errors we assess the adequacy of our approximation and of another approximation with the help of two examples

on the zeros of a class of polynomials including the generalized bessel polynomials abstract it is proved that the real part of any zero of the polynomial pn 1 x of degree n which is defined by an 1pn 1 x anpn 1 x bnpn x xcnpn x p0 x 0 p1 x 1 is negative in the case bn 0 cn 0 a consequence of this result is that the zeros of the bessel polynomials as well as the zeros of the generalized bessel polynomials for a 2 have negative real parts moreover an estimate of the real part which can be easily found improves a well known inequality in the case a 3n 1

decomposition of perturbed chebyshev polynomials we characterize polynomial decomposition f email protected q with r email protected c x of perturbed chebyshev polynomials defined by the recurrencef 0 x b f 1 x x c f n 1 x x d f n x af n 1 x n 1 where a b c email protected r and a 0 these polynomials generalize the chebyshev polynomials which are obtained by setting a 14 c d 0 and email protected 1 2 at the core of the method two algorithms for polynomial decomposition are provided which allow to restrict the investigation to the resolution of six systems of polynomial equations in three variables the final task is then carried out by the successful computation of reduced grobner bases with maple 10 some additional data for the calculations are available on the author s web page

orthogonal polynomials with a constant recursion formula and an application to harmonic analysis on calcule la mesure pour laquelle une suite de polynomes avec une formule de recurrence constante est orthogonale

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

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

the remainder in weyl s law for heisenberg manifolds we prove that the error term r in weyl s law is o 5 6 for certain three dimensional heisenberg manifolds we also show that the l 2 norm of the weyl error term integrated over the moduli space of left invariant heisenberg metrics is 3 4 we conjecture that r o 3 4 i s a sharp deterministic upper bound for heisenberg three manifolds

on co recursive orthogonal polynomials and their application to potential scattering abstract let pn x n 0 be a system of polynomials orthogonal with respect to a positive definite moment functional and satisfying the recurrence relation pn x x cn pn 1 x npn 2 x n 1 2 where p0 x 1 and p 1 x 0 the corresponding co recursive orthogonal polynomials p n x n 0 satisfy the same recurrence relations except for n 1 where now p 1 x x c 1 0 and p 0 x 1 the p n are orthogonal with respect to a moment functional which is positive definite for 0 and quasi definite for p n x separation theorems true interval of orthogonality etc can be determined from those of the pn x these polynomials occur in the l2 solution of the radial schrodinger equation for a separable potential where pn x is the tchebichef polynomial of the second kind in the case of s waves

on isospectral periodic potentials on a discrete lattice i on considere une version discretisee et generalisee de l equation de schrodinger et on etudie le probleme aux valeurs propres

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

riemannian nilmanifolds and the trace formula this paper examines the clean intersection hypothesis required for the expression of the wave invariants computed from the asymptotic expansion of the classical wave trace by duistermaat and guillemin the main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary riemannian two step nilmanifold to satisfy the clean intersection hypothesis this condition is stated in terms of metric lie algebra data we use the calculation to show that generic two step nilmanifolds satisfy the clean intersection hypothesis in contrast we also show that the family of two step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two step nilmanifolds finally we give examples of nilmanifolds that fail the clean intersection hypothesis

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained