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Journal of Computational and Applied Mathematics Extensions of Discrete
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 225 (2009) 440–451 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Extensions of discrete classical orthogonal polynomials beyond the orthogonality R.S. Costas-Santos a,∗, J.F. Sánchez-Lara b a Department of Mathematics, University of California, South Hall, Room 6607 Santa Barbara, CA 93106, USA b Universidad Politécnica de Madrid, Escuela Técnica Superior de Arquitectura, Departamento de Matemática Aplicada, Avda Juan de Herrera, 4. 28040 Madrid, Spain article info a b s t r a c t α,β Article history: It is well-known that the family of Hahn polynomials fhn .xI N/gn≥0 is orthogonal with Received 5 November 2007 respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can MSC: be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for 33C45 Hahn polynomials for a degree higher than N. 42C05 We also present analogous results for dual Hahn, Krawtchouk, and Racah polynomials 34B24 and give the limit relations among them for all n 2 N0. Furthermore, in order to get Keywords: these results for the Krawtchouk polynomials we will obtain a more general property of Classical orthogonal polynomials orthogonality for Meixner polynomials. Inner product involving difference ' 2008 Elsevier B.V. All rights reserved. -
Extensions of Discrete Classical Orthogonal Polynomials Beyond the Orthogonality
W&M ScholarWorks Arts & Sciences Articles Arts and Sciences 2009 Extensions of discrete classical orthogonal polynomials beyond the orthogonality R. S. Costas-Santos J. F. Sanchez-Lara Follow this and additional works at: https://scholarworks.wm.edu/aspubs Recommended Citation Costas-Santos, R. S., & Sánchez-Lara, J. F. (2009). Extensions of discrete classical orthogonal polynomials beyond the orthogonality. Journal of Computational and Applied Mathematics, 225(2), 440-451. This Article is brought to you for free and open access by the Arts and Sciences at W&M ScholarWorks. It has been accepted for inclusion in Arts & Sciences Articles by an authorized administrator of W&M ScholarWorks. For more information, please contact [email protected]. Journal of Computational and Applied Mathematics 225 (2009) 440–451 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Extensions of discrete classical orthogonal polynomials beyond the orthogonality R.S. Costas-Santos a,∗, J.F. Sánchez-Lara b a Department of Mathematics, University of California, South Hall, Room 6607 Santa Barbara, CA 93106, USA b Universidad Politécnica de Madrid, Escuela Técnica Superior de Arquitectura, Departamento de Matemática Aplicada, Avda Juan de Herrera, 4. 28040 Madrid, Spain article info a b s t r a c t α,β Article history: It is well-known that the family of Hahn polynomials fhn .xI N/gn≥0 is orthogonal with Received 5 November 2007 respect to a certain weight function up to degree N. In this paper we prove, by using the three-term recurrence relation which this family satisfies, that the Hahn polynomials can MSC: be characterized by a ∆-Sobolev orthogonality for every n and present a factorization for 33C45 Hahn polynomials for a degree higher than N. -
Image Analysis by Discrete Orthogonal Dual Hahn Moments
Pattern Recognition Letters 28 (2007) 1688–1704 www.elsevier.com/locate/patrec Image analysis by discrete orthogonal dual Hahn moments Hongqing Zhu a, Huazhong Shu a,c,*, Jian Zhou a, Limin Luo a,c, J.L. Coatrieux b,c a Laboratory of Image Science and Technology, Department of Computer Science and Engineering, Southeast University, 210096 Nanjing, People’s Republic of China b INSERM U642, Laboratoire Traitement du Signal et de l’Image, Universite´ de Rennes I, 35042 Rennes, France c Centre de Recherche en Information Biome´dicale Sino-franc¸ais (CRIBs), France Received 29 May 2005; received in revised form 26 February 2007 Available online 29 April 2007 Communicated by L. Younes Abstract In this paper, we introduce a set of discrete orthogonal functions known as dual Hahn polynomials. The Tchebichef and Krawtchouk polynomials are special cases of dual Hahn polynomials. The dual Hahn polynomials are scaled to ensure the numerical stability, thus creating a set of weighted orthonormal dual Hahn polynomials. They are allowed to define a new type of discrete orthogonal moments. The discrete orthogonality of the proposed dual Hahn moments not only ensures the minimal information redundancy, but also elim- inates the need for numerical approximations. The paper also discusses the computational aspects of dual Hahn moments, including the recurrence relation and symmetry properties. Experimental results show that the dual Hahn moments perform better than the Legendre moments, Tchebichef moments, and Krawtchouk moments in terms of image reconstruction capability in both noise-free and noisy con- ditions. The dual Hahn moment invariants are derived using a linear combination of geometric moments. -
The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its Q-Analogue
The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Roelof Koekoek Ren´eF. Swarttouw Abstract We list the so-called Askey-scheme of hypergeometric orthogonal polynomials and we give a q- analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence rela- tion, the second order differential or difference equation, the forward and backward shift operator, the Rodrigues-type formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation, second order difference equation, forward and backward shift operator, Rodrigues-type formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal poly- nomials. Finally, in chapter 5 we point out how the ‘classical’ hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues. Acknowledgement We would like to thank Professor Tom H. Koornwinder who suggested us to write a report like this. He also helped us solving many problems we encountered during the research and provided us with several references. Contents Preface 5 Definitions and miscellaneous formulas 7 0.1 Introduction . 7 0.2 The q-shifted factorials . 8 0.3 The q-gamma function and the q-binomial coefficient . 10 0.4 Hypergeometric and basic hypergeometric functions . -
Notes on the Founding of the DUKE MATHEMATICAL JOURNAL
Notes on the Founding of the DUKE MATHEMATICAL JOURNAL Title page to the bound first volume of DMJ CONTENTS Introduction 1 Planning a New Journal 3 Initial Efforts 3 First Discussion with the AMS 5 Galvanizing Support for a New Journal at Duke 7 Second Discussion with the AMS 13 Regaining Initiative 16 Final Push 20 Arrangements with Duke University Press 21 Launching a New Journal 23 Selecting a Title for the Journal and Forming a Board of Editors 23 Transferring Papers to DMJ 28 Credits 31 Notes 31 RESOLUTION The Council of the American Mathematical Society desires to express to the officers of Duke University, to the members of the Department of Mathematics of the University, and to the other members of the Editorial Board of the Duke Mathematical Journal its grateful appreciation of the service rendered by the Journal to mathematical science, and to extend to all those concerned in its management the congratulations of the Society on the distinguished place which it has assumed from the beginning among the significant mathematical periodicals of the world. December 31, 1935 St. Louis, Missouri INTRODUCTION In January 1936, just weeks after the completion of the first volume of the Duke Mathematical Journal (DMJ), Roland G. D. Richardson, professor and dean of the graduate school at Brown University, longtime secretary of the American Mathematical Society (AMS), and one of the early proponents of the founding of the journal, wrote to Duke President William Preston Few to share the AMS Council’s recent laudatory resolution and to offer his own personal note of congratulations: In my dozen years as Secretary of the American Mathematical Society no project has interested me more than the founding of this new mathematical journal. -
Orthogonal Polynomials and Special Functions
Orthogonal Polynomials and Special Functions SIAM Activity Group on Orthogonal Polynomials and Special Functions ???? Newsletter ???? Published Three Times a Year October 1997 Volume 8, Number 1 Contents formatik, Mathematik und Naturwissenschaften). My new address there is: From the Editor . 1 Prof. Dr. Wolfram Koepf Message from the Chair: Charter Renewal . 2 Fachbereich IMN Reports from Meetings and Conferences . 4 HTWK Leipzig Forthcoming Meetings and Conferences . 5 Gustav-Freytag-Str. 42 A Books and Journals . 9 D-04277 Leipzig Call For Papers . 16 phone: +49-341-307 64 95 Software Announcements . 17 fax: +49-341-301 27 22 Problems and Solutions . 17 e-mail: [email protected] Miscellaneous . 19 WWW: http://www.imn.htwk-leipzig.de/~koepf How to Contribute to the Newsletter . 24 Activity Group: Addresses . 24 At least for the next few months, I will, however, be still a±liated with Konrad-Zuse-Zentrum so that you can send your messages to both places, L P O N A L Y and my old WWW homepage will still be valid. O N G O O M These changes are time-consuming, and I am H I A T L R SIAM indebted to Tom Koornwinder and Martin Mul- S O Activity Group doon who do a great job as Editors of the elec- tronic newsletter OP-SF Net. Fortunately I could S S P E Est. 1990 N borrow much material from them. Indeed, their C O I I A C T material was the main source for my collection L F U N today. I would like to announce that I will resign my From the Editor editorship at the end of the period of my appoint- ment, i.e. -
TABLE of CONTENTS Elementary Problems and Solutions. .Edited By
THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION a TABLE OF CONTENTS On a New Kind of Numbers A.K. Agarwal 194 Some Sequences of Large Integers. Henry Ibstedt 200 Characteristics and the Three Gap Theorem .Tony van Ravenstein, Graham Winley and Keith Tognetti 204 Announcement on Fourth International Conference on Fibonacci Numbers and Their Applications 214 A Two-Variable Lagrange-Type Inversion Formula with Applications to Expansion and Convolution Identities Christian Krattenthaler 215 Lambert Series and the Summation of Reciprocals in Certain Fibonacci-Lucas-Type Sequences R. Andre- Jeannin 223 Third International Conference Proceedings 226 The Distribution of Residues of Two-Term Recurrence Sequences Eliot Jacobson 227 Generalizations of the Dual Zeckendorf Integer Representation Theorems—Discovery by Fibonacci Trees and Word Patterns J.C. Turner and T.D. Robb 230 The Fibonacci Sequence and the Time Complexity of Generating the Conway Polynomial and Related Topological Invariants Phillip G. Bradford 240 An Algebraic Indentity and Some Partial Convolutions W.C. Chu 252 On Certain Number-Theoretic Inequalities J. Sandor and L. Toth 255 Length of the /z-Number Game Anne Ludington-Young 259 Tiling the kth Power of a Power Series .-..../. Arkin, D.C. Arney, G.E. Bergum, S.A. Burr and B.J. Porter 266 On a Hoggatt-Bergum Paper with Totient Function Approach for Divisibility and Congruence Relations Sahib Singh 273 Elementary Problems and Solutions. .Edited by A.P.Hillman 277 Advanced Problems and Solutions Edited by Raymond E. Whitney 283 £ vVoOL UME 28 AUGUST 1990 NUMBER 3 PURPOSE The primary function of THE FIBONACCI QUARTERLY is to serve as a focal point for widespread interest in the Fibonacci and related numbers, especially with respect to new results, research proposals, challenging problems, and innovative proofs of old ideas. -
Bibliography
Bibliography 1. N. ABDUL-HALIM and W.A. AL-SALAM, A characterization of the Laguerre polynomials. Rendiconti del Seminario Matematico della UniversitadiPadova` 34, 1964, 176–179. 2. M. ABRAMOWITZ and I.A. STEGUN (eds.), Handbook of mathematical functions (with for- mulas, graphs, and mathematical tables). Dover Publications, New York, 1970. 3. L.D. ABREU and J. BUSTOZ, Turan´ inequalities for symmetric Askey-Wilson polynomials. The Rocky Mountain Journal of Mathematics 30, 2000, 401–409. 4. S. AHMED,A.LAFORGIA and M.E. MULDOON, On the spacing of the zeros of some classi- cal orthogonal polynomials. Journal of the London Mathematical Society (2) 25, 1982, 246– 252. 5. S. AHMED,M.E.MULDOON and R. SPIGLER, Inequalities and numerical bounds for ze- ros of ultraspherical polynomials. SIAM Journal on Mathematical Analysis 17, 1986, 1000– 1007. 6. K. ALLADI and M.L. ROBINSON, Legendre polynomials and irrationality. Journal fur¨ die Reine und Angewandte Mathematik 318, 1980, 137–155. 7. WM.R. ALLAWAY, The representation of orthogonal polynomials in terms of a differential operator. Journal of Mathematical Analysis and Applications 56, 1976, 288–293. 8. WM.R. ALLAWAY, Some properties of the q-Hermite polynomials. Canadian Journal of Mathematics 32, 1980, 686–694. 9. WM.R. ALLAWAY, Convolution orthogonality and the Jacobi polynomials. Canadian Math- ematical Bulletin 32, 1989, 298–308. 10. WM.R. ALLAWAY, Convolution shift, c-orthogonality preserving maps, and the Laguerre polynomials. Journal of Mathematical Analysis and Applications 157, 1991, 284–299. 11. N.A. AL-SALAM, Orthogonal polynomials of hypergeometric type. Duke Mathematical Jour- nal 33, 1966, 109–121. -
EPADEL a Semisesquicentennial History, 1926-2000
EPADEL A Semisesquicentennial History, 1926-2000 David E. Zitarelli Temple University An MAA Section viewed as a microcosm of the American mathematical community in the twentieth century. Raymond-Reese Book Company Elkins Park PA 19027 Author’s address: David E. Zitarelli Department of Mathematics Temple University Philadelphia, PA 19122 e-mail: [email protected] Copyright © (2001) by Raymond-Reese Book Company, a division of Condor Publishing, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photography, recording, or any information storage retrieval system, without written permission from the publisher. Requests for permission to make copies of any part of the work should be mailed to Permissions, Raymond-Reese Book Company, 307 Waring Road, Elkins Park, PA 19027. Printed in the United States of America EPADEL: A Sesquicentennial History, 1926-2000 ISBN 0-9647077-0-5 Contents Introduction v Preface vii Chapter 1: Background The AMS 1 The Monthly 2 The MAA 3 Sections 4 Chapter 2: Founding Atlantic Apathy 7 The First Community 8 The Philadelphia Story 12 Organizational Meeting 13 Annual Meeting 16 Profiles: A. A. Bennett, H. H. Mitchell, J. B. Reynolds 21 Chapter 3: Establishment, 1926-1932 First Seven Meetings 29 Leaders 30 Organizational Meeting 37 Second Meeting 39 Speakers 40 Profiles: Arnold Dresden, J. R. Kline 48 Chapter 4: Émigrés, 1933-1941 Annual Meetings 53 Leaders 54 Speakers 59 Themes of Lectures 61 Profiles: Hans Rademacher, J. A. Shohat 70 Chapter 5: WWII and its Aftermath, 1942-1955 Annual Meetings 73 Leaders 76 Presenters 83 Themes of Lectures 89 Profiles: J. -
Hypergeometric Orthogonal Polynomials and Their Q-Analogues
Springer Monographs in Mathematics Hypergeometric Orthogonal Polynomials and Their q-Analogues Bearbeitet von Roelof Koekoek, Peter A Lesky, René F Swarttouw, Tom H Koornwinder 1. Auflage 2010. Buch. xix, 578 S. Hardcover ISBN 978 3 642 05013 8 Format (B x L): 15,5 x 23,5 cm Gewicht: 2230 g Weitere Fachgebiete > Mathematik > Mathematische Analysis Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Contents Foreword .......................................................... v Preface ............................................................ xi 1 Definitions and Miscellaneous Formulas ........................... 1 1.1 Orthogonal Polynomials . 1 1.2 The Gamma and Beta Function . 3 1.3 The Shifted Factorial and Binomial Coefficients . 4 1.4 Hypergeometric Functions . 5 1.5 The Binomial Theorem and Other Summation Formulas . 7 1.6 SomeIntegrals............................................. 8 1.7 TransformationFormulas.................................... 10 1.8 The q-ShiftedFactorial...................................... 11 1.9 The q-Gamma Function and q-Binomial Coefficients . 13 1.10 Basic Hypergeometric Functions . 15 1.11 The q-Binomial Theorem and Other Summation Formulas -
The Askey-Scheme of Hypergeometric Orthogonal Polynomials And
The Askeyscheme of hyp ergeometric orthogonal p olynomials and its q analogue Ro elof Ko eko ek Rene F Swarttouw February Abstract We list the socalled Askeyscheme of hyp ergeometric orthogonal p olynomials In chapter we give the denition the orthogonality relation the three term recurrence relation and generating functions of all classes of orthogonal p olynomials in this scheme In chapter we give all limit relations b etween dierent classes of orthogonal p olynomials listed in the Askeyscheme In chapter we list the q analogues of the p olynomials in the Askeyscheme We give their denition orthogonality relation three term recurrence relation and generating functions In chapter we give the limit relations b etween those basic hyp ergeometric orthogonal p olynomials Finally in chapter we p oint out how the classical hyp ergeometric orthogonal p olynomials of the Askeyscheme can b e obtained from their q analogues Acknowledgement We would like to thank Professor Tom H Ko ornwinder who suggested us to write a rep ort like this He also help ed us solving many problems we encountered during the research and provided us with several references Contents Preface Denitions and miscellaneous formulas Intro duction The q shifted factorials The q gamma function and the q binomial co ecient Hyp ergeometric and basic hyp ergeometric functions The q binomial theorem and other summation formulas Transformation -
The Different Tongues of Q-Calculus
Proceedings of the Estonian Academy of Sciences, 2008, 57, 2, 81–99 doi: 10.3176/proc.2008.2.03 Available online at www.eap.ee/proceedings The different tongues of q-calculus Thomas Ernst Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden; [email protected] Received 14 December 2007, in revised form 7 February 2008 Abstract. In this review paper we summarize the various dialects of q-calculus: quantum calculus, time scales, and partitions. The close connection between Gq(x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notation will be illustrated by the (q-)Euler reflection formula, q-Appell functions, Carlitz– AlSalam polynomials, and the so-called q-addition. We conclude with some short biographies about famous scientists in q-calculus. Key words: elliptic functions, theta functions, q-Appell functions, q-addition, Carlitz–AlSalam polynomial. 1. INTRODUCTION q-Calculus is a generalization of many subjects, like hypergeometric series, complex analysis, and particle physics. In short, q-calculus is quite a popular subject today. It has developed various dialects like quantum calculus, time scales, partitions, and continued fractions. We will give several aspects of q-calculus to enlighten the strong interdisciplinary as well as mathematical character of this subject. The close connection between q-calculus on the one hand, and elliptic functions and theta functions on the other hand will be shown. Some new formulas will be given, such as the two restatements of Kummer’s 2F1(¡1) formula (in q-form) (36) and (37).