Orthogonal polynomial solutions of linear ordinary dierential equations

An electronic timepiece includes a reversible stepping motor to drive rotatable hands to provide a time display. In a first preferred embodiment, the electronic timepiece includes a circuit means to generate alternating current pulses having an increased pulse width or an increased amplitude to drive the stepping motor with an increased driving current during high speed time correction when a manually operable external control member is actuated during time correction. In a second preferred embodiment, the electronic timepiece comprises a clockwise correction switch, a counter-clockwise correction switch and circuit means for generating first and second alternating current pulses of first and second pulse widths at a predetermined frequency higher than normal driving pulses to drive a stepping motor in clockwise and counter-clockwise directions, respectively, to perform clockwise and counter-clockwise corrections when the clockwise and counter-clockwise correction switches are actuated during time corrections, respectively. The pulse width of the second alternating current pulses is selected to be smaller than that of the normal driving pulses.

[1]  A. Zayed.  Advances in Shannon's Sampling Theory , 1993 .

[2]  On Sobolev Orthogonality for the Generalized Laguerre Polynomials , 1996 .

[3]  Finding differential equations for symmetric generalized ultraspherical polynomials by using inversion methods , 1999, math/9908147.

[4]  Kil Hyun Kwon,et al.  Characterizations of orthogonal polynomials satisfying differential equations , 1994 .

[5]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[6]  Luc Haine,et al.  Bispectral darboux transformations: An extension of the Krall polynomials , 1997 .

[7]  H. Bavinck.  Differential operators having Sobolev type Laguerre polynomials as eigenfunctions , 2001 .

[8]  F. A. Grünbaum,et al.  Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation , 1996 .

[9]  W. N. Everitt,et al.  The symmetric form of the Koekoeks' Laguerre type differential equation , 1995 .

[10]  H. Bavinck,et al.  Differential operators having symmetric orthogonal polynomials as eigenfunctions , 1999 .

[11]  R. Koekoek,et al.  On a difference equation for generalizations of Charlier polynomials , 1995, math/9908144.

[12]  F. Alberto Grünbaum,et al.  Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials , 1998 .

[13]  Sobolev orthogonality for the Gegenbauer polynomials {C n (-N+ 1 2 ) } n?0 , 1998 .

[14]  Mourad E. H. Ismail,et al.  An electrostatics model for zeros of general orthogonal polynomials , 2000 .

[15]  Jesús S. Dehesa,et al.  Some connection and linearization problems for polynomials in and beyond the Askey scheme , 2001 .

[16]  L. Littlejohn,et al.  Orthogonal polynomial eigenfunctions of second-order partial differerential equations , 2001 .

[17]  R. Koekoek.  Differential equations for symmetric generalized ultraspherical polynomials , 1999, math/9908143.

[18]  I. Jung,et al.  Differential equations and Sobolev orthogonality , 1995 .

[19]  Andrei Martínez-Finkelshtein,et al.  Asymptotic properties of Sobolev orthogonal polynomials , 1998 .

[20]  P. Forrester,et al.  Electrostatics and the zeros of the classical polynomials , 1986 .

[21]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[22]  Kil Hyun Kwon,et al.  CLASSIFICATION OF CLASSICAL ORTHOGONAL POLYNOMIALS , 1997 .

[23]  E. C. Titchmarsh.  ON EXPANSIONS IN EIGENFUNCTIONS (II) , 1941 .

[24]  R. Koekoek.  Generalizations of the classical laguerre polynomials and some q-analogues , 1990 .

[25]  Wolfgang Hahn,et al.  Über die Jacobischen Polynome und zwei verwandte Polynomklassen , 1935 .

[26]  H. L. Krall,et al.  Differential equations of infinite order for orthogonal polynomials , 1966 .

[27]  The jacobi inversion formula , 1999, math/9908148.

[28]  K. Kwon,et al.  A characterization of Hermite polynomials , 1997 .

[29]  Antonio J. Durán,et al.  Functions with Given Moments and Weight Functions for Orthogonal Polynomials , 1993 .

[30]  Francisco Marcellán,et al.  On recurrence relations for Sobolev orthogonal polynomials , 1995 .

[31]  H. L. Krall,et al.  Certain differential equations for Tchebycheff polynomials , 1938 .

[32]  Lance L. Littlejohn,et al.  Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture , 2001, J. Approx. Theory.

[33]  F. Marcellán,et al.  Jacobi-Sobolev-type orthogonal polynomials: second-order differential equation and zeros , 1998 .

[34]  H. Bavinck,et al.  Differential and difference operators having orthogonal polynomials with two linear perturbations as eigenfunctions , 1998 .

[35]  H. Bavinck.  A note on the Koekoek's differential equation for generalized Jacobi polynomials , 2000 .

[36]  F. Marcellán,et al.  Orthogonal polynomials on Sobolev spaces: old and new directions , 1993 .

[37]  F. Grünbaum,et al.  Differential equations in the spectral parameter , 1986 .

[38]  H. Bavinck.  Differential operators having Sobolev-type Jacobi polynomials as eigenfunctions , 2003 .

[39]  D. Nualart,et al.  Chaotic and predictable representation for L'evy Processes , 2000 .

[40]  M. Ismail.  More on Electrostatic Models for Zeros of Orthagonal Polynomials , 2000 .

[41]  Allan M. Krall,et al.  Orthogonal polynomials satisfying fourth order differential equations , 1981, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[42]  F. Grünbaum.  Electrostatic interpretation for the zeros of certain polynomials and the Darboux process , 2001 .

[43]  E. Hendriksen,et al.  A Bessel type orthogonal polynomial system , 1984 .

[44]  Kil Hyun Kwon,et al.  Orthogonalizing Weights of Tchebychev Sets of Polynomials , 1992 .

[45]  Pascal Maroni.  An integral representation for the Bessel form , 1995 .

[46]  S. Bochner,et al.  Über Sturm-Liouvillesche Polynomsysteme , 1929 .

[47]  H. Bavinck.  A direct approach to Koekoek's differential equation for generalized Laguerre polynomials , 1995 .

[48]  Alexei Zhedanov,et al.  A method of constructing Krall's polynomials , 1999 .

[49]  H. Kramer,et al.  A Generalized Sampling Theorem , 1959 .

[50]  An application in stochastics of the Laguerre-type polynomials , 2001 .

[51]  L. Vinet,et al.  A method to study the Krall and q -Krall polynomials , 2001 .

[52]  R. Koekoek,et al.  A generalization of Laguerre polynomials , 1993 .

[53]  A. Sri Ranga,et al.  Orthogonal Functions , 1998 .

[54]  H. Bavinck.  Linear perturbations of differential or difference operators with polynomials as eigenfunctions , 1997 .

[55]  Tom H. Koornwinder,et al.  Orthogonal Polynomials With Weight Function (1 - x) α ( l + x) β + M δ(x + 1) + Nδ(x - 1) , 1984, Canadian Mathematical Bulletin.

[56]  F. Grünbaum,et al.  The q -version of a theorem of Bochner , 1996 .

[57]  Lance L. Littlejohn,et al.  Symmetric and Symmetrisable Differential Expressions , 1990 .

[58]  Sobolev orthogonal polynomials: The discrete-continuous case , 1999 .

[59]  J. Koekoek,et al.  Differential equations for generalized Jacobi polynomials , 2000 .

[60]  I. Jung,et al.  Sobolev orthogonal polynomials and spectral differential equations , 1995 .

[61]  Antonio J. Durán,et al.  The Stieltjes moments problem for rapidly decreasing functions , 1989 .

[62]  Lance L. Littlejohn,et al.  A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations , 2002 .

[63]  F. Alberto Grünbaum,et al.  Some functions that generalize the Krall-Laguerre polynomials , 1999 .

[64]  Roelof Koekoek,et al.  On a differential equation for Koornwinder's generalized Laguerre polynomials , 1991 .

[65]  Gangjoon Yoon,et al.  Symmetrizability of differential equations having orthogonal polynomial solutions , 1997 .

[66]  Kil Hyun Kwon,et al.  Self-Adjoint Operators Generated from Non-Lagrangian Symmetric Differential Equations Having Orthogonal Polynomial Eigenfunctions , 2001 .

[67]  Lance L. Littlejohn,et al.  On the classification of differential equations having orthogonal polynomial solutions , 1984 .

[68]  K. H. Kwon,et al.  SOBOLEV ORTHOGONAL POLYNOMIALS AND SECOND ORDER DIFFERENTIAL EQUATION II , 1996 .

[69]  Peter Lesky,et al.  Die charakterisierung der klassischen orthogonalen polynome durch Sturm-Liouvillesche Differentialgleichungen , 1962 .

[70]  P. Butzer,et al.  Sampling theorems associated with fourth- and higher-order self-adjoint eigenvalue problems , 1994 .