A Necessary Condition for a Nontrivial Zero of the Riemann Zeta Function Via the Polylogarithmic Function Lazhar Fekih-Ahmed
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A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions
axioms Article A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions Maxie D. Schmidt School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30318, USA; [email protected] or [email protected] Received: 23 April 2019; Accepted: 17 May 2019; Published: 19 May 2019 Abstract: The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way. Keywords: generating function; series transformation; gamma function; Hankel contour MSC: 05A15; 30E20; 31B10; 11B73 1. Introduction 1.1. Definitions Given a sequence f fngn≥0, we adopt the notation for the respective ordinary generating function (OGF), F(z), and exponential generating function (EGF), Fb(z), of the sequence in some formal indeterminate parameter z 2 C: n F(z) = ∑ fnz (1) n≥0 fn Fb(z) = ∑ zn. n≥0 n! Notice that we can always construct these functions over any sequence f fngn2N and formally perform operations on these functions within the ring of formal power series in z without any considerations on the constraints imposed by the convergence of the underlying series as a complex function of z. -
Zeta Functions and Basic Analogues
Zeta Functions and Basic Analogues by Peter John Anderson B.Sc. (Hon.), Simon Fraser University, 1985 M.Sc., Simon Fraser University, 1991 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mathematics and Statistics c Peter John Anderson, 2009 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l’édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-66824-5 Our file Notre référence ISBN: 978-0-494-66824-5 NOTICE: AVIS: The author has granted a non- L’auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l’Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distribute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L’auteur conserve la propriété du droit d’auteur ownership and moral rights in this et des droits moraux qui protège cette thèse. -
Fractional Calculus and Generalized Mittag-Leffler Type Functions
Fractional calculus and generalized Mittag-Leffler type functions Christian Lavault∗ Abstract In this paper, the generalized fractional integral operators of two generalized Mittag- Leffler type functions are investigated. The special cases of interest involve the generalized Fox–Wright function and the generalized M-series and K-function. In the next Section 2 we first recall some generalized fractional integral operators among the most widely used in fractional calculus. Section 3 is devoted to the definitions of M- series and K-function and their relations to special functions. In Sections 4 and 5, effective fractional calculus of the generalized M-series and the K-function is carried out. The last section briefly concludes and opens up new perspectives. The results established herein generalize recent properties of generalized Mittag-Leffler type functions using left- and right-sided generalized fractional differintegral operators. The note results also in important applications in physics and mathematical engineering. Keywords: Fox–Wright psi function; Generalized hypergeometric function; M-series and K-function; Mittag-Leffler type functions; Riemann–Liouville’s, Saigo’s and Saigo–Maeda’s generalized fractional calculus operators. 2010 Mathematics Subject Classification: 26A33, 33C05, 33C10, 33C20, 33C60, 44A15. Contents 1 Introduction and motivations 1 1.1 The Mittag-Leffler and generalized Mittag-Leffler type functions .............. 1 1.2 Integral representations of Mittag-Leffler type functions ................... 6 1.3 Higher transcendental functions . .............. 8 2 Fractional calculus of generalized M-L type functions 13 2.1 Riemann–Liouville fractional calculus . ................ 13 2.2 Saigo’s fractional differintegration operators . .................... 14 2.3 Saigo–Maeda’s fractional differintegration operators ..................... 16 arXiv:1703.01912v2 [math.CA] 21 Mar 2017 3 The generalized M-series and K-function 17 3.1 Relations to the Fox–Wright and M-L type functions . -
Identities for the Gamma and Hypergeometric Functions: an Overview
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Wits Institutional Repository on DSPACE Identities for the gamma and hypergeometric functions: an overview from Euler to the present Julie Patricia Hannah School of Mathematics University of the Witwatersrand Johannesburg South Africa Under the supervision of Professor S. J. Johnston and Dr. S. Currie A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013 i Declaration I declare that this Dissertation is my own, unaided work. It is being submitted for the Degree of Masters of Science at the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination at any other university. Signed: Julie Hannah day of , 2013 in Johannesburg ii Abstract Equations involving the gamma and hypergeometric functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions to differential and integral equations. In this work we trace a brief history of the development of the gamma and hypergeometric functions, illustrate the close relationship between them and present a range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Our literature review will show that while continued research into hypergeometric identities has generated many new results, some of these can be shown to be variations of known identities. Hence, we will also discuss computer based methods that have been developed for creating and analysing such identities, in order to check for originality and for numerical validity. -
On the Theory of Zeta-Functions and L-Functions
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2015 On the Theory of Zeta-functions and L-functions Almuatazbellah Awan University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Awan, Almuatazbellah, "On the Theory of Zeta-functions and L-functions" (2015). Electronic Theses and Dissertations, 2004-2019. 53. https://stars.library.ucf.edu/etd/53 ON THE THEORY OF ZETA-FUNCTIONS AND L-FUNCTIONS by ALMUATAZBELLAH AWAN B.S. University of Central Florida, 2012 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Spring Term 2015 Major Professor: Ram Mohapatra c 2015 Almuatazbellah Awan ii ABSTRACT In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its gener- alizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet’s L-function. We have also investi- gated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Num- ber Theorem and the Riemann Hypothesis. -
ANALYTIC COMBINATORICS — COMPLEX ASYMPTOTICS (Chapters IV, V, VI, VII)
ANALYTIC COMBINATORICS — COMPLEX ASYMPTOTICS (Chapters IV, V, VI, VII) PHILIPPE FLAJOLET & ROBERT SEDGEWICK Algorithms Project Department of Computer Science INRIA Rocquencourt Princeton University 78153 Le Chesnay Princeton, NJ 08540 France USA Zeroth Edition, May 8, 2004 (This temporary version expires on December 31, 2004) ~ ~ ~ ~ ~ ~ i ABSTRACT This booklet develops in about 240 pages the basics of asymptotic enumeration through an approach that revolves around generating functions and complex analysis. Major proper- ties of generating functions that are of interest here are singularities. The text presents the core of the theory with two chapters on complex analytic methods focusing on rational and meromorphic functions as well as two chapters on fundamentals of singularity analysis and combinatorial consequences. It is largely oriented towards applications of complex anal- ysis to asymptotic enumeration and asymptotic properties of random discrete structures. Many examples are given that relate to words, integer compositions, paths and walks in graphs, lattice paths, constrained permutations, trees, mappings, walks, and maps. Acknowledgements. This work was supported in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). This booklet would be substantially different (and much less informative) without Neil Sloane’s Encyclopedia of Integer Sequences and Steve Finch’s Mathematical Constants, both available on the internet. Bruno Salvy and Paul Zimmermann have developed algorithms and libraries for combinatorial structures and generating functions that are based on the MAPLE system for symbolic computations and have proven to be immensely useful. This is a set of lecture notes that are a component of a wider book project titled Analytic Combi- natorics, which will provide a unified treatment of analytic methods in combinatorics. -
Methods of Theoretical Physics 614
Methods of Theoretical Physics 614 ABSTRACT Second part of the Fall 2009 course 614, on Mathematical Methods in Theoretical Physics. This partial set of notes begins with complex analysis. Contents 1 Functions of a Complex Variable 2 1.1 Complex Numbers, Quaternions and Octonions . ...... 2 1.2 Analytic or Holomorphic Functions . ..... 13 1.3 ContourIntegration .............................. 19 1.4 Classification of Singularities . ...... 31 1.5 TheOppenheimFormula ............................ 45 1.6 CalculusofResidues .............................. 49 1.7 Evaluation of real integrals . 50 1.8 SummationofSeries ............................... 62 1.9 AnalyticContinuation . 64 1.10TheGammaFunction .............................. 68 1.11 TheRiemannZetaFunction. 75 1.12 AsymptoticExpansions . 84 1.13 MethodofSteepestDescent . 88 1 1 Functions of a Complex Variable 1.1 Complex Numbers, Quaternions and Octonions The extension from the real number system to complex numbers is an important one both within mathematics itself, and also in physics. The most obvious area of physics where they are indispensable is quantum mechanics, where the wave function is an intrinsically complex object. In mathematics their use is very widespread. One very important point is that by generalising from the real to the complex numbers, it becomes possible to treat the solution of polynomial equations in a uniform manner, since now not only equations like x2 1 =0 but also x2 + 1 = 0 can be solved. − The complex numbers can be defined in terms of ordered pairs of real numbers. Thus we may define the complex number z to be the ordered pair z = (x,y), where x and y are real. Of course this doesn’t tell us much until we give some rules for how these quantities behave.