DOCSLIB.ORG

6-3.Pdf, 298 Kb

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
6-3.Pdf, 298 Kb Orthogonal Polynomials and Special Functions SIAM Activity Group on Orthogonal Polynomials and Special Functions ???? Newsletter ???? Published Three Times a Year June 1996 Volume 6, Number 3 Contents of members of our Activity Group. In particu- lar, both the Chair, Charles Dunkl, and the Vice From the Editor . 1 Chair, Tom H. Koornwinder, now have their own Memorial Note about Waleed Al-Salam . 2 WWW home pages whose addresses are listed on Memorial Note about Lawrence C. Biedenharn . 3 page 19. Reports from Meetings and Conferences . 4 Forthcoming Meetings and Conferences . 6 I invite everybody to consult some of the Books and Journals . 12 WWW pages which can give interesting infor- Software Announcements . 15 Problems and Solutions . 15 mation about orthogonal polynomials and special Miscellaneous . 18 functions. Rather than looking a formula up in a How to Contribute to the Newsletter . 18 mathematical dictionary, the desired information Activity Group: Addresses . 19 could be online available to you! Steven Finch is developing a WWW page on L P O N A L Y Favorite Mathematical Constants and their inter- O N G O O M relations which was announced in the last issue H I A T of the Newsletter. His site includes a wealth of L R SIAM S O mathematical knowledge and can be visited at Activity Group http://www.mathsoft.com/cgi-shl/constant.bat. S S P E Est. 1990 N C O At http://www.cecm.sfu.ca/projects/ISC/ you I I A C T L F U N have access to the Inverse Symbolic Calculator that looks up in a huge database to identify deci- mal numbers and uses modern algorithmic meth- ods to generate algebraic identities between sev- From the Editor eral of them. Using these methods, the formula µ ¶ µ ¶ The electronic media, and, in particular, the X1 4 2 1 1 1 k ¼ = ¡ ¡ ¡ World Wide Web, influence the daily work of re- 8k + 1 8k + 4 8k + 5 8k + 6 16 searchers all over the world more and more. This k=0 is reflected, for example, in the continually in- was discovered recently; it gives the opportunity creasing number of personal WWW home pages to calculate far out coe±cients of ¼'s hexadecimal June 1996 Orthogonal Polynomials and Special Functions Newsletter 2 SIAM Activity Group Wilson scheme, equipped with any standardiza- on tion you may wish to use; see also p. 15. Im- Orthogonal Polynomials and Special Functions plementations of the editor are used for this 3 purpose. This interface can be visited at Elected O±cers http://www.can.nl/~renes/CAOP.html. Charles Dunkl, Chair You see there is a lot to discover! Strolling Tom H. Koornwinder, Vice Chair through these pages will furthermore link you to Willard Miller, Program Director many other interesting sites. I hope you all can Nico M. Temme, Secretary take advantage of the possibilities which these new Appointed O±cers media o®er! Wolfram Koepf, Editor of the Newsletter If you know about interesting WWW pages con- Martin E. Muldoon, Webmaster cerning orthogonal polynomials and special func- 3 tions that I did not mention, please submit the The purpose of the Activity Group is corresponding addresses to the Editor. Our mem- |to promote basic research in orthogonal polyno- bers should receive such information. mials and special functions; to further the application of this subject in other parts of mathematics, and in May 31, 1996 Wolfram Koepf science and industry; and to encourage and support the exchange of information, ideas, and techniques between workers in this ¯eld, and other mathemati- cians and scientists. Memorial Note about Waleed Al-Salam 1926-1996 Friends and colleagues are saddened by news of expansion without the computation of early ones, the passing, a few months short of his 70th birth- using single precision arithmetic! day, of Waleed Al-Salam, Professor Emeritus of Mathematics at the University of Alberta. Born Victor Adamchik and Stan Wagon present a on July 15, 1926 in Baghdad, Iraq, he died on deduction of a generalization of the above formula April 14, 1996 in Edmonton, Alberta, Canada. using symbolic computation on their WWW page http://www.wri.com/~victor/articles/pi/pi.html. Waleed studied at the University of California, Berkeley, earning a Bachelor's degree in Engineer- David H. Bailey and Simon Plou®e who ing Physics in 1950 and a M.A. in Mathematics have discovered the above representation to- in 1951. He returned to Baghdad as an Instructor gether with Peter Borwein have their own at the College of Science for a few years, before WWW page on Recognizing Real Numbers at enrolling for the Ph.D. at Duke University. He http://www.cecm.sfu.ca/organics/papers/bailey/. completed the degree in 1958 with a thesis On the An archive on Numbers with details about Bessel Polynomials written under the supervision history, theory, applications, as well as printed of Leonard Carlitz. By this time he was already and electronic references can be found at a regular contributor to the periodical literature http://archives.math.utk.edu/subjects/numbers.html with some 20 published articles on a variety of containing e.g., a link to continued fractions. topics in orthogonal polynomials and special func- Concerning families of orthogonal polynomi- tions. als, Ren¶e Swarttouw designed an interface to After completing his Ph.D., Waleed returned Maple which currently enables the generation to the College of Science in Baghdad as Asso- of plots as well as an online computation ciate Professor. Coming back to North America of recurrence, di®erential and di®erence equa- in 1962, he eventually (1966) took a position at tions for orthogonal polynomials of the Askey- the University of Alberta, where he was Professor June 1996 Orthogonal Polynomials and Special Functions Newsletter 3 of Mathematics from 1967 until his retirement in Memorial Note about 1992. Waleed continued to contribute to several Lawrence C. Biedenharn areas related to orthogonal polynomials; his CV lists over 80 articles. Areas covered by his work in- Lawrence C. Biedenharn Jr., 73, a longtime clude characterization theorems (see his survey ar- member of Duke University's physics faculty and ticle in pp. 1{24 of P. Nevai, ed., Orthogonal Poly- an internationally known researcher in theoretical nomials: Theory and Practice, Kluwer, 1990), physics, died Monday, Feb. 12, in Austin, Texas, Turan expressions, generating functions, summa- after a lengthy battle with cancer. He had made tion formulas, q-analogs, and fractional operators. his home in Texas in recent years. He and his collaborators did much work on vari- ous special and generalized systems of orthogonal Biedenharn became the youngest full professor polynomials. The Al-Salam-Carlitz polynomials on the Duke faculty|at age 38|when he was ap- (1965) are still frequently cited; see, e.g., the pa- pointed in 1961. He remained at Duke until 1993, pers of R. Askey and S.K. Suslov in Lett. Math. when he retired as James B. Duke professor of Phys. 29 (1993), 123-132 and J. Phys. A 20 (1993), physics and subsequently moved to the University L693-L698. The Al-Salam-Chihara polynomials of Texas at Austin as adjunct professor. (1976) play an important role in the Askey-Wilson A native of Vicksburg, Miss., he received both scheme of basic hypergeometric orthogonal poly- his bachelor's degree and his Ph.D. from the Mas- nomials. sachusetts Institute of Technology and served in Waleed supervised the Ph.D. work of Bill All- World War II as a Signal Corps o±cer and later away (1972) and Mourad Ismail (1974), and con- on the sta® of Gen. Douglas MacArthur in Japan. ducted joint work with a variety of people at Al- Serving on the faculties of Yale University and berta and elsewhere. In later years, these in- Rice University before coming to Duke, he pub- cluded Ted Chihara, A. Verma, Mourad Ismail lished six books and hundreds of research articles and Waleed's wife, Nadhla Al-Salam, also on the in the ¯elds of nuclear physics and later math- faculty at Alberta, and a member of this Activity ematical physics. He also edited the Journal of Group. Mathematical Physics for many years. On his retirement in 1992, Waleed decided to Biedenharn, Holman, and Louck showed how put his expertise and energy at the disposal of the the classical work on ordinary hypergeometric se- orthogonal polynomials community by starting ries is intimately related to the irreducible rep- and maintaining an ftp site for papers in the area. resentations of the compact group SU(2). Simi- This e®ort prospered and he continued to oversee larly, they found U(n) multiple series generaliza- it until last year when his failing health made it tions of one-variable hypergeometric summation necessary to pass the task to Hans Haubold at the and transformation theorems by comparing two UN O±ce in Vienna. Waleed had been diagnosed ways of computing the matrix elements of multi- with leukemia in 1993 and this was to reduce his plicity free Wigner and Racah coe±cients in U(n). active participation in conferences in subsequent This work was done in the context of the quan- years. Nevertheless, with Nadhla's constant sup- tum theory of angular momentum and the spe- port, he still managed to travel and old and new cial unitary groups SU(n). This work motivated friends were able to bene¯t from his knowledge the far-reaching q-analogs that Milne, Gustafson, and enjoy his optimistic and humorous personal- and their co-workers subsequently found. Appli- ity. cations of this later work include uni¯ed proofs Martin Muldoon of the Macdonald identities, constant term iden- ([email protected]) tities, multiple q-beta integrals, U(n + 1) and symplectic generalizations of the Bailey Trans- form and Bailey Lemma, classical matrix inver- sion results, numerous classical summation and June 1996 Orthogonal Polynomials and Special Functions Newsletter 4 transformation theorems for one-variable q-series, high computational power of this approach by giving Rogers-Ramanujan identities, and, ¯nally, new in- one-line proofs of properties of the Bernoulli polyno- ¯nite families of identities for sums of squares in mials.
Load more

Recommended publications
  • Curriculum Vita
    CURRICULUM VITA NAME: Mourad E.H. Ismail AFFILIATION: University of Central Florida and King Saud University and ADDRESS: Department of Mathematics University of Central Florida 4000 Central Florida Blvd. P.O. Box 161364 Orlando, FL 32816-1364, USA. Telephones: Cell (407) 770-9959. POSITION: Research Professor at U of Central Florida PERSONAL DATA: Born April 27, 1944, in Cairo, Egypt. Male. Married. Canadian and Egyptian citizen, Permanent resident in the United States. EDUCATION: Ph.D. 1974 (Alberta), M.Sc. 1969 (Alberta), B.Sc. 1964 (Cairo). MATHEMATICAL GENEALOGY: Source: The Mathematics Genealogy Project, http://www.genealogy.ams.org • Yours truly, Ph. D. 1974, Advisor: Waleed Al-Salam. • Waleed Al-Salam, Ph. D. 1958, Advisor: Leonard Carlitz. • Leonard Carlitz, Ph. D. 1930, Advisor: Howard Mitchell. • Howard Mitchell, Ph. D. 1910, Advisor: Oswald Veblen. • Oswald Veblen, Ph. D. 1903, Advisor: Eliakim Hastings Moore. • Eliakim Hastings Moore, Ph. D. 1885, Advisor: H. A. Newton. • H. A. Newton, B. S. Yale 1850, Advisor: Michel Chasles. • Michel Chasles, Ph. D. 1814, Advisor: Simeon Poisson. • Simeon Poisson, Advisor: Joseph Lagrange. • Joseph Lagrange, Ph. D. Advisor: Leonhard Euler. 1 RESEARCH INTERESTS: Approximation theory, asymptotics, combinatorics, integral transforms and operational calculus, math- ematical physics, orthogonal polynomials and special functions. POSITIONS HELD: 2010{2012 Chair Professor, City University of Hong Kong 2003-2012 and 2013-present Professor, University of Central Florida 2008{2016 Distinguished Research
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • THE ANNUAL MEETING in MIAMI the Seventieth Annual Meeting Of
    THE ANNUAL MEETING IN MIAMI The seventieth Annual Meeting of the American Mathematical Society was held at the University of Miami, Coral Gables, and Mi­ ami, Florida, on January 23-27, 1964, in conjunction with meetings of the Mathematical Association of America. All sessions were held at the University of Miami with the exception of the Gibbs Lecture, which was held at the Everglades Hotel. Registration at the meeting was 1493, including 1196 members of the Society. The thirty-seventh Josiah Willard Gibbs Lecture was delivered by Professor Lars Onsager of Yale University at 8:00 P.M. on Friday, January 24, 1964. His lecture was entitled Mathematical problems of cooperative phenomena. President Doob presided. Professor Deane Montgomery of the Institute for Advanced Study delivered his retiring Presidential Address, Compact groups of trans­ formations at 9:00 A.M. on Friday, January 24. Professor Montgom­ ery was introduced by Professor R. L. Wilder. By invitation of the Committee to Select Hour Speakers for Sum­ mer and Annual Meetings, hour addresses were given by Professor Morton Brown, of the University of Michigan, at 9:00 A.M. Thurs­ day, January 23, and Professor Heisuke Hironaka of Brandeis Uni­ versity of 2:00 P.M., Friday, January 24. Professor Brown, who was introduced by Professor J. H. Curtiss, spoke on Topological manifolds. Professor Hironaka spoke on Singularities in algebraic varieties and was introduced by Professor R. D. James. There were four special sessions of invited twenty-minute papers as follows: in geometry, organized by Herbert Busemann and with speakers Herbert Busemann, Louis Auslander, R.
    [Show full text]
  • Enumerative and Algebraic Combinatorics in the 1960'S And
    Enumerative and Algebraic Combinatorics in the 1960's and 1970's Richard P. Stanley University of Miami (version of 17 June 2021) The period 1960{1979 was an exciting time for enumerative and alge- braic combinatorics (EAC). During this period EAC was transformed into an independent subject which is even stronger and more active today. I will not attempt a comprehensive analysis of the development of EAC but rather focus on persons and topics that were relevant to my own career. Thus the discussion will be partly autobiographical. There were certainly deep and important results in EAC before 1960. Work related to tree enumeration (including the Matrix-Tree theorem), parti- tions of integers (in particular, the Rogers-Ramanujan identities), the Redfield- P´olya theory of enumeration under group action, and especially the repre- sentation theory of the symmetric group, GL(n; C) and some related groups, featuring work by Georg Frobenius (1849{1917), Alfred Young (1873{1940), and Issai Schur (1875{1941), are some highlights. Much of this work was not concerned with combinatorics per se; rather, combinatorics was the nat- ural context for its development. For readers interested in the development of EAC, as well as combinatorics in general, prior to 1960, see Biggs [14], Knuth [77, §7.2.1.7], Stein [147], and Wilson and Watkins [153]. Before 1960 there are just a handful of mathematicians who did a sub- stantial amount of enumerative combinatorics. The most important and influential of these is Percy Alexander MacMahon (1854-1929). He was a highly original pioneer, whose work was not properly appreciated during his lifetime except for his contributions to invariant theory and integer parti- tions.
    [Show full text]
  • 8. A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." the Fibonacci Quarterly 3.3 (1965): 161-76
    EVALUATION OF CERTAIN INFINITE SERIES INVOLVING TERMS OF GENERALIZED SEQUENCES 8. A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 3.3 (1965): 161-76. 9. A. F. Horadam & P. Filipponi. "Cholesky Algorithm Matrices of Fibonacci Type and Proper- ties of Generalized Sequences." The Fibonacci Quarterly 29.2 (1991): 164-73. 10. M. V. Koutras. "Eulerian Numbers Associated with Sequences of Polynomials." The Fibo- nacci Quarterly 32.1 (1994):44-57. 11. R. S. Melham & A. G. Shannon. "On Reciprocal Sums of Chebyshev Related Sequences." The Fibonacci Quarterly 33.3 (1995): 194-202. 12. G.J. Tee. "Integer Sums of Recurring Series." New Zealand1 of'Math. 22 (1993):85-100. 13. I. M. Vinogradov. Elements of Number Theory. New York: Dover, 1954. 14. Problem B-758. The Fibonacci Quarterly 32.1 (1994):86. AMS Classification Numbers: 11B75, 11B39, 11L03 IN MEMORIAM—LEONARD CARLITZ Leonard Carlitz, a long-time friend and supporter of The Fibonacci Association, passed away on September 17, 1999. For many years Carlitz was on the editorial board of The Fibonacci Quarterly, and between 1963 and 1984 he published 72 articles in the Quarterly (including 19 joint papers and 7 short notes). Carlitz was born in 1907 in Philadelphia, and he grew up in that city. He won a scholarship to the University of Pennsylvania where he completed his AB degree in 1927, his MA degree in 1928, and his Ph.D. in 1930—all in mathematics. His Ph.D. thesis advisor was H. H. Mitchell, who had been a student of Oswald Veblen who, in turn, had studied under E.
    [Show full text]
  • IN MEMORIAM Leonard Carlitz (1907I1999)
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Finite Fields and Their Applications 6, 203d206 (2000) doi.10.1006/!ta.2000.0276, available online at http://www.idealibrary.com on IN MEMORIAM Leonard Carlitz (1907I1999) Leonard Carlitz, one of the most proli"c mathematical researchers of all time, died in the early morning hours of September 17, 1999, in Pittsburgh, Pennsylvania, following a period of declining health. He was 91 years old. Carlitz was preceded in death by his wife, Clara, and two sisters, Ruth Katz and Rosalyn Gross. He is survived by two sons, Michael Carlitz of Palo Alto, California, a retired IBM computer programmer, and Robert D. Carlitz of Pittsburgh, Pennsylvania, a University of Pittsburgh physics professor who also runs the nonpro"t organization Information Renaissance. Also surviv- ing are granddaughters Natasha Carlitz of Menlo Park, California, and Ruth Denali Carlitz of Pittsburgh, Pennsylvania, and nieces Myrna Guest of London, England, and Naomi Lazard of East Hampton, New York. During the years 1930}1990, Carlitz published a total of 770 papers in many areas of mathematics including combinatorics, number theory, "eld theory and polynomials, commutative rings and algebras, algebraic geo- metry, linear algebra, special functions, "nite di!erences, and geometry. Another paper [5] appeared in 1995 in a special issue of Finite Fields and ¹heir Applications which was dedicated to him, but the research for that last paper was done many years earlier. He was the sole author of 668 of these 771 papers; the remaining 103 papers were written with 39 di!erent co-authors.
    [Show full text]
  • [Fim = (M-Am)/2, So That
    EVALUATION O F CERTAIN INFINITE SERIES INVOLVING T E R M S O F GENERALIZED SEQUENCES Piero Filipponi Fondazione Ugo Bordoni, Via B. Castiglione 59,1-00142, Rome, Italy e-mail: [email protected] (Submitted September 1998-Final Revision March 1999) 1. INTRODUCTION AND PRELIMINARIES This article deals with the generalized Fibonacci numbers Uk(m) and the generalized Lucas numbers Vk(m) which have already been considered in [9], [11], and elsewhere. These numbers satisfy the second-order recurrence relation Wk(m) = mWk_l(m) + Wk_2(m) (k>2) (1.1) where W stands for either U or V, and m is an arbitrary integer. The initial conditions in (1.1) are W0(m) - 0, Wx(m) = 1, or WQ(m) = 2, Wx{m) - m depending on whether Wis Uor V. Whenever no misunderstanding can arise, Uk(m) and Vk(m) will be denoted simply by Uk and Vk, respectively. Closed-form expressions (Binet forms) for these numbers are k k \Uk{m) = (a m-p m)lhm, k k \yk{m) = a m+p m, where lam = (m + Am)/2, (1.3) [fim = (m-Am)/2, so that <*mPm = -\ am+pm=m, and am-fim = Am. (1.4) It can be proved that the extension through negative values of the subscript leads to j ^ - 1 ' " " " (1.5) ih Observe that Uk(V) = Fk and Vk(l) = Lk (the k Fibonacci and Lucas number, respectively), ih while £4(2) = Pk and Vk(2) - Qk (the k Pell and Pell-Lucas number, respectively). The principal aim of this paper is to generalize [14], and some results established in [1], [4], and [12] (see also [5]), by finding an explicit form for the infinite series where Vk obviously stands for Vk(m) and h, r, and s are positive integers, the last two of which are subject to the restriction, rls<llam.
    [Show full text]
  • Crm Proceedings & Lecture Notes
    Volume 22 C CRM R PROCEEDINGS & M LECTURE NOTES Centre de Recherches Mathématiques Montréal Algebraic Methods and q-Special Functions Jan Felipe van Diejen Luc Vinet Editors American Mathematical Society Algebraic Methods and q-Special Functions https://doi.org/10.1090/crmp/022 Volume 22 C CRM R PROCEEDINGS & M LECTURE NOTES Centre de Recherches Mathématiques Montréal Algebraic Methods and q-Special Functions Jan Felipe van Diejen Luc Vinet Editors The Centre de Recherches Mathématiques (CRM) of the Université de Montréal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Université de Montréal, the Province of Québec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathématiques (ISM) of Montréal, whose constituent members are Concordia University, McGill University, the Université de Montréal, the Université de Québec à Montréal, and the Ecole Polytechnique. The CRM may be reached on the Web at www.crm.umontreal.ca. American Mathematical Society Providence, Rhode Island USA The production of this volume was supported in part by the Fonds pour la Formation de Chercheurs et l’Aide `alaRecherche(FondsFCAR)andtheNaturalSciencesand Engineering Research Council of Canada (NSERC). 1991 Mathematics Subject Classification.Primary33D45;Secondary05E05,33C50, 33C80, 43A85, 43A90, 81R05, 81R10. Library of Congress Cataloging-in-Publication Data Algebraic methods and q-special functions / Jan Felipe van Diejen, Luc Vinet, editors. p. cm. — (CRM proceedings & lecture notes, ISSN 1065-8580 ; v. 22) Includes bibliographical references.
    [Show full text]
  • 2016 Newsletter
    2016 DEPARTMENT OF MATHEMATICS UN IVERSITY of WASHINGTON 1 This newsletter is published annually for alumni & friends of Mathematics at the University of Washington I’ve got sunshine on a cloudy day. Half a century ago, the Temptations’ David Ruffin opened the nation’s number one good fortune that the cost is low. Little or no equipment is required. One inevitably song with that line. I happily sang along, even if the source of that sunshine, “My thinks in this regard of the Indian mathematical genius Ramanujan, largely self Girl,” was not yet on the horizon. After all, I did have mathematics, and for me, taught, who had the good luck of choosing to write to the English mathematician as for mathematicians around the world, this indeed makes every day sunny, for G.H. Hardy in 1913. Hardy was so taken by Ramanujan’s novel ideas that he had we inhabit a world of wondrous objects and ideas. We are blessed. Ramanujan come to Cambridge the next year to collaborate. (For more on this, see the recent movie The Man Who Knew Infinity or read Robert Kanigel’s book on With this blessing comes responsibility, the responsibility to share our luminous which it was based.) Ramanujan was unique, but the larger lesson is that potential world with others and strive to ensure that anyone with interest and motivation mathematicians are everywhere. It is our responsibility not just to teach those who has access to it. Members of our department do this daily, through our teaching. come to our door, but also to provide opportunities for those less visible.
    [Show full text]