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Generalizations of Euler Numbers and Polynomials 1
GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS QIU-MING LUO AND FENG QI Abstract. In this paper, the concepts of Euler numbers and Euler polyno- mials are generalized, and some basic properties are investigated. 1. Introduction It is well-known that the Euler numbers and polynomials can be defined by the following definitions. Definition 1.1 ([1]). The Euler numbers Ek are defined by the following expansion t ∞ 2e X Ek = tk, |t| ≤ π. (1.1) e2t + 1 k! k=0 In [4, p. 5], the Euler numbers is defined by t/2 ∞ n 2n 2e t X (−1) En t = sech = , |t| ≤ π. (1.2) et + 1 2 (2n)! 2 n=0 Definition 1.2 ([1, 4]). The Euler polynomials Ek(x) for x ∈ R are defined by xt ∞ 2e X Ek(x) = tk, |t| ≤ π. (1.3) et + 1 k! k=0 It can also be shown that the polynomials Ei(t), i ∈ N, are uniquely determined by the following two properties 0 Ei(t) = iEi−1(t),E0(t) = 1; (1.4) i Ei(t + 1) + Ei(t) = 2t . (1.5) 2000 Mathematics Subject Classification. 11B68. Key words and phrases. Euler numbers, Euler polynomials, generalization. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China. This paper was typeset using AMS-LATEX. 1 2 Q.-M. LUO AND F. QI Euler polynomials are related to the Bernoulli numbers. For information about Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4]. -
An Identity for Generalized Bernoulli Polynomials
1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.11.2 47 6 23 11 An Identity for Generalized Bernoulli Polynomials Redha Chellal1 and Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria [email protected] [email protected] [email protected] Mohamed Mehbali Centre for Research Informed Teaching London South Bank University London United Kingdom [email protected] Abstract Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli poly- nomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identi- ties for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling. 1Corresponding author. 1 1 Introduction Let N and C denote, respectively, the set of positive integers and the set of complex numbers. (α) In his book, Roman [41, p. 93] defined generalized Bernoulli polynomials Bn (x) as follows: for all n ∈ N and α ∈ C, we have ∞ tn t α B(α)(x) = etx. (1) n n! et − 1 Xn=0 The Bernoulli numbers Bn, classical Bernoulli polynomials Bn(x), and generalized Bernoulli (α) numbers Bn are, respectively, defined by (1) (α) (α) Bn = Bn(0), Bn(x)= Bn (x), and Bn = Bn (0). (2) The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory, mathematical analysis, and topology. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis -
Higher Order Bernoulli and Euler Numbers
David Vella, Skidmore College [email protected] Generating Functions and Exponential Generating Functions • Given a sequence {푎푛} we can associate to it two functions determined by power series: • Its (ordinary) generating function is ∞ 풏 푓 풙 = 풂풏풙 풏=ퟏ • Its exponential generating function is ∞ 풂 품 풙 = 풏 풙풏 풏! 풏=ퟏ Examples • The o.g.f and the e.g.f of {1,1,1,1,...} are: 1 • f(x) = 1 + 푥 + 푥2 + 푥3 + ⋯ = , and 1−푥 푥 푥2 푥3 • g(x) = 1 + + + + ⋯ = 푒푥, respectively. 1! 2! 3! The second one explains the name... Operations on the functions correspond to manipulations on the sequence. For example, adding two sequences corresponds to adding the ogf’s, while to shift the index of a sequence, we multiply the ogf by x, or differentiate the egf. Thus, the functions provide a convenient way of studying the sequences. Here are a few more famous examples: Bernoulli & Euler Numbers • The Bernoulli Numbers Bn are defined by the following egf: x Bn n x x e 1 n1 n! • The Euler Numbers En are defined by the following egf: x 2e En n Sech(x) 2x x e 1 n0 n! Catalan and Bell Numbers • The Catalan Numbers Cn are known to have the ogf: ∞ 푛 1 − 1 − 4푥 2 퐶 푥 = 퐶푛푥 = = 2푥 1 + 1 − 4푥 푛=1 • Let Sn denote the number of different ways of partitioning a set with n elements into nonempty subsets. It is called a Bell number. It is known to have the egf: ∞ 푆 푥 푛 푥푛 = 푒 푒 −1 푛! 푛=1 Higher Order Bernoulli and Euler Numbers th w • The n Bernoulli Number of order w, B n is defined for positive integer w by: w x B w n xn x e 1 n1 n! th w • The n Euler Number of -
Problem Books in Mathematics
Problem Books in Mathematics Edited by K.A. Bencsath P.R. Halmos Springer-Science+Business Media, LLC Problem Books in Mathematics Series Editors: K.A. Bencsath and P.R. Halmos Pell's Equation by Edward J. Barbeau Polynomials by Edward J. Barbeau Problems in Geometry by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint-Raymond Problem Book for First Year Calculus by George W Bluman Exercises in Probability by T. Cacoullos Probability Through Problems by Marek Capinski and Tomasz Zastawniak An Introduction to Hilbert Space and Quantum Logic by David W Cohen Unsolved Problems in Geometry by Ballard T. Croft, Kenneth J. Falconer, and Richard K Guy Berkeley Problems in Mathematics, (Third Edition) by Paulo Ney de Souza and Jorge-Nuno Silva Problem-Solving Strategies by Arthur Engel Problems in Analysis by Bernard R. Gelbaum Problems in Real and Complex Analysis by Bernard R. Gelbaum Theorems and Counterexamples in Mathematics by Bernard R. Gelbaum and John MH. Olmsted Exercises in Integration by Claude George (continued after index) Richard K. Guy Unsolved Problems in Number Theory Third Edition With 18 Figures , Springer Richard K. Guy Department of Mathematics and Statistics University of Calgary 2500 University Drive NW Calgary, Alberta Canada T2N IN4 [email protected] Series Editors: Katalin A. Bencsath Paul R. Halmos Mathematics Department of Mathematics School of Science Santa Clara University Manhattan College Santa Clara, CA 95053 Riverdale, NY 10471 USA USA [email protected] [email protected] Mathematics Subject Classification (2000): OOA07, 11-01, 11-02 Library of Congress Cataloging-in-Publication Data Guy, Richard K. -
Handbook of Number Theory Ii
HANDBOOK OF NUMBER THEORY II by J. Sändor Babes-Bolyai University ofCluj Department of Mathematics and Computer Science Cluj-Napoca, Romania and B. Crstici formerly the Technical University of Timisoara Timisoara Romania KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON Contents PREFACE 7 BASIC SYMBOLS 9 BASIC NOTATIONS 10 1 PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 15 1.1 Introduction 15 1.2 Some historical facts 16 1.3 Even perfect numbers 20 1.4 Odd perfect numbers 23 1.5 Perfect, multiperfect and multiply perfect numbers 32 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers 36 1.7 Superperfect and related numbers 38 1.8 Pseudoperfect, weird and harmonic numbers 42 1.9 Unitary, bi-unitary, infinitary-perfect and related numbers 45 1.10 Hyperperfect, exponentially perfect, integer-perfect and y-perfect numbers 50 1.11 Multiplicatively perfect numbers 55 1.12 Practical numbers 58 1.13 Amicable numbers 60 1.14 Sociable numbers 72 References 77 2 GENERALIZATIONS AND EXTENSIONS OF THE MÖBIUS FUNCTION 99 2.1 Introduction 99 1 CONTENTS 2.2 Möbius functions generated by arithmetical products (or convolutions) 106 1 Möbius functions defined by Dirichlet products 106 2 Unitary Möbius functions 110 3 Bi-unitary Möbius function 111 4 Möbius functions generated by regulär convolutions .... 112 5 K-convolutions and Möbius functions. B convolution . ... 114 6 Exponential Möbius functions 117 7 l.c.m.-product (von Sterneck-Lehmer) 119 8 Golomb-Guerin convolution and Möbius function 121 9 max-product (Lehmer-Buschman) 122 10 Infinitary -
Fall 2008 [Pdf]
Le Bulletin du CRM • crm.math.ca • Automne/Fall 2008 | Volume 14 – No 2 | Le Centre de recherches mathématiques The Fall 2008 Aisenstadt Chairs Four Aisenstadt Chair lecturers will visit the CRM during the 2008-2009 thematic year “Probabilistic Methods in Mathemat- ical Physics.” We report here on the series of lectures of Wendelin Werner (Université Paris-Sud 11) and Andrei Okounkov (Princeton University), both of whom are Fields Medalists, who visited the CRM in August and September 2008 respectively. The other Aisenstadt Chairs will be held by Svante Janson (Uppsala University) and Craig Tracy (University of California at Davis). Wendelin Werner Andrei Okounkov de Yvan Saint-Aubin (Université de Montréal) by John Harnad (Concordia University) Wendelin Werner est un A metaphor from an ancient fragment by Archilochus “The Fox spécialiste de la théo- knows many things but the Hedgehog knows one big thing” rie des probabilités. Il a was used by Isaiah Berlin as title and theme of his essay on Tol- obtenu son doctorat en stoy’s view of history (“. by nature a fox, but believed in be- 1993 sous la direction de ing a hedgehog”) [1]. It is also very suitably applied to styles in Jean-François Le Gall. Il science. In his presentation of the work of Andrei Okounkov est professeur au labo- when he was awarded the Fields medal at the 2006 Interna- ratoire de mathématiques tional Congress of Mathematicians in Madrid, Giovanni Felder à l’Université Paris-Sud said: “Andrei Okounkov’s initial area of research was group XI à Orsay depuis 1997, representation theory, with particular emphasis on combinato- ainsi qu’à l’École nor- rial and asymptotic aspects. -
Mathematical Constants and Sequences
Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) .. -
MASON V(Irtual) Mid-Atlantic Seminar on Numbers March 27–28, 2021
MASON V(irtual) Mid-Atlantic Seminar On Numbers March 27{28, 2021 Abstracts Amod Agashe, Florida State University A generalization of Kronecker's first limit formula with application to zeta functions of number fields The classical Kronecker's first limit formula gives the polar and constant term in the Laurent expansion of a certain two variable Eisenstein series, which in turn gives the polar and constant term in the Laurent expansion of the zeta function of a quadratic imaginary field. We will recall this formula and give its generalization to more general Eisenstein series and to zeta functions of arbitrary number fields. Max Alekseyev, George Washington University Enumeration of Payphone Permutations People's desire for privacy drives many aspects of their social behavior. One such aspect can be seen at rows of payphones, where people often pick an available payphone most distant from already occupied ones.Assuming that there are n payphones in a row and that n people pick payphones one after another as privately as possible, the resulting assignment of people to payphones defines a permutation, which we will refer to as a payphone permutation. It can be easily seen that not every permutation can be obtained this way. In the present study, we consider different variations of payphone permutations and enumerate them. Kisan Bhoi, Sambalpur University Narayana numbers as sum of two repdigits Repdigits are natural numbers formed by the repetition of a single digit. Diophantine equations involving repdigits and the terms of linear recurrence sequences have been well studied in literature. In this talk we consider Narayana's cows sequence which is a third order liner recurrence sequence originated from a herd of cows and calves problem. -
The Geometry Center Reaches
THE NEWSLETTER OF THE MA THEMA TICAL ASSOCIATION OF AMERICA The Geometry Center Reaches Out Volume IS, Number 1 A major mission of the NSF-sponsored University of Minnesota Geometry Center is to support, develop, and promote the communication of mathematics at all levels. Last year, the center increased its efforts to reach and to educate diffe rent and dive rse groups ofpeople about the beauty and utility of mathematics. Center members Harvey Keynes and Frederick J. Wicklin describe In this Issue some recent efforts to reach the general public, professional mathematicians, high school teach ers, talented youth, and underrepresented groups in mathematics. 3 MAA President's Column Museum Mathematics Just a few years ago, a trip to the local 7 Mathematics science museum resembled a visit to Awareness Week a taxidermy shop. The halls of the science museum displayed birds of 8 Highlights from prey, bears, cougars, and moose-all stiff, stuffed, mounted on pedestals, the Joint and accompanied by "Don't Touch" Mathematics signs. The exhibits conveyed to all Meetings visitors that science was rigid, bor ing, and hardly accessible to the gen 14 NewGRE eral public. Mathematical Fortunately times have changed. To Reasoning Test day even small science museums lit erally snap, crackle, and pop with The graphical interface to a museum exhibit that allows visitors to interactive demonstrations of the explore regular polyhedra and symmetries. 20 Letters to the physics of electricity, light, and sound. Editor Visitors are encouraged to pedal, pump, and very young children to adults, so it is accessible push their way through the exhibit hall. -
NEWSLETTER No
NEWSLETTER No. 458 May 2016 NEXT DIRECTOR OF THE ISAAC NEWTON INSTITUTE In October 2016 David Abrahams will succeed John Toland as Director of the Isaac Newton Institute for Mathematical Sciences and NM Rothschild and Sons Professor of Mathematics in Cambridge. David, who is a Royal Society Wolfson Research Merit Award holder, has been Beyer Professor of Applied Mathematics at the University of Man- chester since 1998. From 2014-16 he was Scientific Director of the International Centre for Math- ematical Sciences in Edinburgh and was President of the Institute of Mathematics and its Applica- tions from 2007-2009. David’s research has been in the broad area of applied mathematics, mainly focused on the theoretical understanding of wave processes including scattering, diffraction, localisation and homogenisation. In recent years his research has broadened somewhat, to now cover topics as diverse as mathematical finance, nonlinear vis- He has also been involved in a range of public coelasticity and glaciology. He has close links with engagement activities over the years. He a number of industrial partners. regularly offers mathematics talks of interest David plays an active role within the internation- to school students and the general public, al mathematics community, having served on over and ran the annual Meet the Mathematicians 30 national and international working parties, outreach events for sixth form students with panels and committees over the past decade. Chris Howls (Southampton). With Chris Budd This has included as a Member of the Applied (Bath) he has organised a training conference Mathematics sub-panel for the 2008 Research As- in 2010 on How to Talk Maths in Public, and in sessment Exercise and Deputy Chair for the Math- 2014 co-chaired the inaugural Festival of Math- ematics sub-panel in the 2014 Research Excellence ematics and its Applications.