Notes Relating to Newton Series for the Riemann Zeta Function
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Asymptotic Estimates of Stirling Numbers
Asymptotic Estimates of Stirling Numbers By N. M. Temme New asymptotic estimates are given of the Stirling numbers s~mJ and (S)~ml, of first and second kind, respectively, as n tends to infinity. The approxima tions are uniformly valid with respect to the second parameter m. 1. Introduction The Stirling numbers of the first and second kind, denoted by s~ml and (S~~m>, respectively, are defined through the generating functions n x(x-l)···(x-n+l) = [. S~m)Xm, ( 1.1) m= 0 n L @~mlx(x -1) ··· (x - m + 1), ( 1.2) m= 0 where the left-hand side of (1.1) has the value 1 if n = 0; similarly, for the factors in the right-hand side of (1.2) if m = 0. This gives the 'boundary values' Furthermore, it is convenient to agree on s~ml = @~ml= 0 if m > n. The Stirling numbers are integers; apart from the above mentioned zero values, the numbers of the second kind are positive; those of the first kind have the sign of ( - l)n + m. Address for correspondence: Professor N. M. Temme, CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands. e-mail: [email protected]. STUDIES IN APPLIED MATHEMATICS 89:233-243 (1993) 233 Copyright © 1993 by the Massachusetts Institute of Technology Published by Elsevier Science Publishing Co., Inc. 0022-2526 /93 /$6.00 234 N. M. Temme Alternative generating functions are [ln(x+1)r :x n " s(ln)~ ( 1.3) m! £..,, n n!' n=m ( 1.4) The Stirling numbers play an important role in difference calculus, combina torics, and probability theory. -
Asymptotics of Singularly Perturbed Volterra Type Integro-Differential Equation
Konuralp Journal of Mathematics, 8 (2) (2020) 365-369 Konuralp Journal of Mathematics Journal Homepage: www.dergipark.gov.tr/konuralpjournalmath e-ISSN: 2147-625X Asymptotics of Singularly Perturbed Volterra Type Integro-Differential Equation Fatih Say1 1Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey Abstract This paper addresses the asymptotic behaviors of a linear Volterra type integro-differential equation. We study a singular Volterra integro equation in the limiting case of a small parameter with proper choices of the unknown functions in the equation. We show the effectiveness of the asymptotic perturbation expansions with an instructive model equation by the methods in superasymptotics. The methods used in this study are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations involving small (or large) parameters. Keywords: Singular perturbation, Volterra integro-differential equations, asymptotic analysis, singularity 2010 Mathematics Subject Classification: 41A60; 45M05; 34E15 1. Introduction A systematic approach to approximation theory can find in the subject of asymptotic analysis which deals with the study of problems in the appropriate limiting regimes. Approximations of some solutions of differential equations, usually containing a small parameter e, are essential in the analysis. The subject has made tremendous growth in recent years and has a vast literature. Developed techniques of asymptotics are successfully applied to problems including the classical long-standing problems in mathematics, physics, fluid me- chanics, astrodynamics, engineering and many diverse fields, for instance, see [1, 2, 3, 4, 5, 6, 7]. There exists a plethora of examples of asymptotics which includes a wide variety of problems in rich results from the very practical to the highly theoretical analysis. -
An Introduction to Asymptotic Analysis Simon JA Malham
An introduction to asymptotic analysis Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter 1. Order notation 5 Chapter 2. Perturbation methods 9 2.1. Regular perturbation problems 9 2.2. Singular perturbation problems 15 Chapter 3. Asymptotic series 21 3.1. Asymptotic vs convergent series 21 3.2. Asymptotic expansions 25 3.3. Properties of asymptotic expansions 26 3.4. Asymptotic expansions of integrals 29 Chapter 4. Laplace integrals 31 4.1. Laplace's method 32 4.2. Watson's lemma 36 Chapter 5. Method of stationary phase 39 Chapter 6. Method of steepest descents 43 Bibliography 49 Appendix A. Notes 51 A.1. Remainder theorem 51 A.2. Taylor series for functions of more than one variable 51 A.3. How to determine the expansion sequence 52 A.4. How to find a suitable rescaling 52 Appendix B. Exam formula sheet 55 3 CHAPTER 1 Order notation The symbols , o and , were first used by E. Landau and P. Du Bois- Reymond and areOdefined as∼ follows. Suppose f(z) and g(z) are functions of the continuous complex variable z defined on some domain C and possess D ⊂ limits as z z0 in . Then we define the following shorthand notation for the relative!propertiesD of these functions in the limit z z . ! 0 Asymptotically bounded: f(z) = (g(z)) as z z ; O ! 0 means that: there exists constants K 0 and δ > 0 such that, for 0 < z z < δ, ≥ j − 0j f(z) K g(z) : j j ≤ j j We say that f(z) is asymptotically bounded by g(z) in magnitude as z z0, or more colloquially, and we say that f(z) is of `order big O' of g(z). -
A Note on the Asymptotic Expansion of the Lerch's Transcendent Arxiv
A note on the asymptotic expansion of the Lerch’s transcendent Xing Shi Cai∗1 and José L. Lópezy2 1Department of Mathematics, Uppsala University, Uppsala, Sweden 2Departamento de Estadística, Matemáticas e Informática and INAMAT, Universidad Pública de Navarra, Pamplona, Spain June 5, 2018 Abstract In [7], the authors derived an asymptotic expansion of the Lerch’s transcendent Φ(z; s; a) for large jaj, valid for <a > 0, <s > 0 and z 2 C n [1; 1). In this paper we study the special case z ≥ 1 not covered in [7], deriving a complete asymptotic expansion of the Lerch’s transcendent Φ(z; s; a) for z > 1 and <s > 0 as <a goes to infinity. We also show that when a is a positive integer, this expansion is convergent for Pm n s <z ≥ 1. As a corollary, we get a full asymptotic expansion for the sum n=1 z =n for fixed z > 1 as m ! 1. Some numerical results show the accuracy of the approximation. 1 Introduction The Lerch’s transcendent (Hurwitz-Lerch zeta function) [2, §25.14(i)] is defined by means of the power series 1 X zn Φ(z; s; a) = ; a 6= 0; −1; −2;:::; (a + n)s n=0 on the domain jzj < 1 for any s 2 C or jzj ≤ 1 for <s > 1. For other values of the variables arXiv:1806.01122v1 [math.CV] 4 Jun 2018 z; s; a, the function Φ(z; s; a) is defined by analytic continuation. In particular [7], 1 Z 1 xs−1e−ax Φ(z; s; a) = −x dx; <a > 0; z 2 C n [1; 1) and <s > 0: (1) Γ(s) 0 1 − ze ∗This work is supported by the Knut and Alice Wallenberg Foundation and the Ministerio de Economía y Competitividad of the spanish government (MTM2017-83490-P). -
NM Temme 1. Introduction the Incomplete Gamma Functions Are Defined by the Integrals 7(A,*)
Methods and Applications of Analysis © 1996 International Press 3 (3) 1996, pp. 335-344 ISSN 1073-2772 UNIFORM ASYMPTOTICS FOR THE INCOMPLETE GAMMA FUNCTIONS STARTING FROM NEGATIVE VALUES OF THE PARAMETERS N. M. Temme ABSTRACT. We consider the asymptotic behavior of the incomplete gamma func- tions 7(—a, —z) and r(—a, —z) as a —► oo. Uniform expansions are needed to describe the transition area z ~ a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for 7(0, z) and V(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given. 1. Introduction The incomplete gamma functions are defined by the integrals 7(a,*)= / T-Vcft, r(a,s)= / t^e^dt, (1.1) where a and z are complex parameters and ta takes its principal value. For 7(0,, z), we need the condition ^Ra > 0; for r(a, z), we assume that |arg2:| < TT. Analytic continuation can be based on these integrals or on series representations of 7(0,2). We have 7(0, z) + r(a, z) = T(a). Another important function is defined by 7>>*) = S7(a,s) = =^ fu^e—du. (1.2) This function is a single-valued entire function of both a and z and is real for pos- itive and negative values of a and z. For r(a,z), we have the additional integral representation e-z poo -zt j.-a r(a, z) = — r / dt, ^a < 1, ^z > 0, (1.3) L [i — a) JQ t ■+■ 1 which can be verified by differentiating the right-hand side with respect to z. -
Harmonic Number Identities Via Polynomials with R-Lah Coefficients
Comptes Rendus Mathématique Levent Kargın and Mümün Can Harmonic number identities via polynomials with r-Lah coeYcients Volume 358, issue 5 (2020), p. 535-550. <https://doi.org/10.5802/crmath.53> © Académie des sciences, Paris and the authors, 2020. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Les Comptes Rendus. Mathématique sont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Comptes Rendus Mathématique 2020, 358, nO 5, p. 535-550 https://doi.org/10.5802/crmath.53 Number Theory / Théorie des nombres Harmonic number identities via polynomials with r-Lah coeYcients Identités sur les nombres harmonique via des polynômes à coeYcients r-Lah , a a Levent Kargın¤ and Mümün Can a Department of Mathematics, Akdeniz University, Antalya, Turkey. E-mails: [email protected], [email protected]. Abstract. In this paper, polynomials whose coeYcients involve r -Lah numbers are used to evaluate several summation formulae involving binomial coeYcients, Stirling numbers, harmonic or hyperharmonic num- bers. Moreover, skew-hyperharmonic number is introduced and its basic properties are investigated. Résumé. Dans cet article, des polynômes à coeYcients faisant intervenir les nombres r -Lah sont utilisés pour établir plusieurs formules de sommation en fonction des coeYcients binomiaux, des nombres de Stirling et des nombres harmoniques ou hyper-harmoniques. De plus, nous introduisons le nombre asymétrique- hyper-harmonique et nous étudions ses propriétés de base. 2020 Mathematics Subject Classification. 11B75, 11B68, 47E05, 11B73, 11B83. Manuscript received 5th February 2020, revised 18th April 2020, accepted 19th April 2020. -
Covering Systems, Kubert Identities and Difference Equations
Mathematica Slovaca Štefan Porubský Covering systems, Kubert identities and difference equations Mathematica Slovaca, Vol. 50 (2000), No. 4, 381--413 Persistent URL: http://dml.cz/dmlcz/136785 Terms of use: © Mathematical Institute of the Slovak Academy of Sciences, 2000 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz Mathematlca Slovaca ©2000 Mathematical Institute Math. Slovaca, 50 (2000), No. 4, 381-413 Slovák Academy of Sciences With the deepest admiration to my esteemed friend Kdlmdn Gyory on the occasion of his sixties COVERING SYSTEMS, KUBERT IDENTITIES AND DIFFERENCE EQUATIONS STEFAN PORUBSKY (Communicated by Stanislav Jakubec ) ABSTRACT. A. S. Fraenkel proved that the following identities involving Ber noulli polynomials m • ч -»n(0) for all n > 0 are true if and only if the system of arithmetic congruences {a^ (mod 6^) : 1 < i < m} is an exact cover of Z. Generalizations of this result involving other functions and more general covering systems have been successively found by A. S. Fraenkel, J. Beebee, Z.-W. Sun and the author. Z.-W. Sun proved an alge braic characterization of functions capable of identities of this type, and indepen dently J. Beebee observed a connection of these results to the Raabe multiplica tion formula for Bernoulli polynomials and with the so-called Kubert identities. -
Physics 112 Mathematical Notes Winter 2000 1. Power Series
Physics 112 Mathematical Notes Winter 2000 1. Power Series Expansion of the Fermi-Dirac Integral The Fermi-Dirac integral is defined as: ∞ 1 xn−1 dx fn(z) ≡ , Γ(n) z−1ex +1 0 µ/kT where x ≡ /kT and z ≡ e . We wish to derive a power series expansion for fn(z)that is useful in the limit of z → 0. We manipulate fn(z) as follows: ∞ z xn−1 dx fn(z)= Γ(n) ex + z 0 ∞ z xn−1 dx = Γ(n) ex(1 + ze−x) 0 ∞ ∞ z −x n−1 m −x m = e x dx (−1) (ze ) Γ(n) 0 m=0 ∞ ∞ z = (−1)mzm e−(m+1)xxn−1 dx . Γ(n) m=0 0 Using the well known integral: ∞ Γ(n) e−Axxn−1 dx = , An 0 and changing the summation variable to p = m + 1, we end up with: ∞ (−1)p−1zp fn(z)= n . (1) p=1 p which is the desired power series expansion. This result is also useful for deriving additional properties ofthe fn(z). For example, it is a simple matter use eq. (1) to verify: ∂ fn−1(z)=z fn(z) . ∂z Note that the classical limit is obtained for z 1. In this limit, which also corresponds to the high temperature limit, fn(z) z. One additional special case is noteworthy. If 1 z =1,weobtain ∞ (−1)p−1 fn(1) = n . p=0 p To evaluate this sum, we rewrite it as follows: 1 − 1 fn(1) = n n p odd p p even p 1 1 − 1 = n + n 2 n p odd p p even p p even p ∞ ∞ 1 − 1 = n 2 n p=1 p p=1 (2p) ∞ − 1 1 = 1 n−1 n . -
A Few Remarks on Values of Hurwitz Zeta Function at Natural and Rational Arguments
Mathematica Aeterna, Vol. 5, 2015, no. 2, 383 - 394 A few remarks on values of Hurwitz Zeta function at natural and rational arguments Pawe÷J. Szab÷owski Department of Mathematics and Information Sciences, Warsaw University of Technology ul. Koszykowa 75, 00-662 Warsaw, Poland Abstract We exploit some properties of the Hurwitz zeta function (n, x) in order to 1 n study sums of the form n j1= 1/(jk + l) and 1 j n 1 n j1= ( 1) /(jk + l) for 2 n, k N, and integer l k/2. We show 1 P ≤ 2 ≤ that these sums are algebraic numbers. We also show that 1 < n N and P n n 2 p Q (0, 1) : the numbers ((n, p) + ( 1) (n, 1 p))/π are algebraic. 2 \ On the way we find polynomials sm and cm of order respectively 2m + 1 and 2m + 2 such that their n th coeffi cients of sine and cosine Fourier transforms are equal to ( 1)n/n2m+1and ( 1)n/n2m+2 respectively. Mathematics Subject Classification: Primary 11M35, 11M06, Sec- ondary 11M36, 11J72 Keywords: Riemann zeta function, Hurwitz zeta function, Bernoulli num- bers, Dirichlet series 1 Introduction Firstly we find polynomials sm and cm of orders respectively 2m+1 and 2m+2 such that their n th coeffi cients in respectively sine and cosine Fourier series are of the form (1)n/n2m+1 and ( 1)n/n2m+2. Secondly using these polyno- mials we study sums of the form: 1 1 1 ( 1)j S(n, k, l) = , S^(n, k, l) = . -
The Gamma Function
The Gamma Function JAMES BONNAR Ψ TREASURE TROVE OF MATHEMATICS February 2017 Copyright ©2017 by James Bonnar. All rights reserved worldwide under the Berne convention and the World Intellectual Property Organization Copyright Treaty. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, ex- cept as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appro- priate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the ac- curacy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. 1 1 + x2=(b + 1)2 1 + x2=(b + 2)2 2 2 × 2 2 × · · · dx ˆ0 1 + x =(a) 1 + x =(a + 1) p π Γ(a + 1 )Γ(b + 1)Γ(b − a + 1 ) = × 2 2 2 Γ(a)Γ(b + 1=2)Γ(b − a + 1) for 0 < a < b + 1=2. -
NOTES on HEAT KERNEL ASYMPTOTICS Contents 1
NOTES ON HEAT KERNEL ASYMPTOTICS DANIEL GRIESER Abstract. These are informal notes on how one can prove the existence and asymptotics of the heat kernel on a compact Riemannian manifold with bound- ary. The method differs from many treatments in that neither pseudodiffer- ential operators nor normal coordinates are used; rather, the heat kernel is constructed directly, using only a first (Euclidean) approximation and a Neu- mann (Volterra) series that removes the errors. Contents 1. Introduction 2 2. Manifolds without boundary 5 2.1. Outline of the proof 5 2.2. The heat calculus 6 2.3. The Volterra series 10 2.4. Proof of the Theorem of Minakshisundaram-Pleijel 12 2.5. More remarks: Formulas, generalizations etc. 12 3. Manifolds with boundary 16 3.1. The boundary heat calculus 17 3.2. The Volterra series 24 3.3. Construction of the heat kernel 24 References 26 Date: August 13, 2004. 1 2 DANIEL GRIESER 1. Introduction The theorem of Minakshisundaram-Pleijel on the asymptotics of the heat kernel states: Theorem 1.1. Let M be a compact Riemannian manifold without boundary. Then there is a unique heat kernel, that is, a function K ∈ C∞((0, ∞) × M × M) satisfying (∂t − ∆x)K(t, x, y) = 0, (1) lim K(t, x, y) = δy(x). t→0+ For each x ∈ M there is a complete asymptotic expansion −n/2 2 (2) K(t, x, x) ∼ t (a0(x) + a1(x)t + a2(x)t + ... ), t → 0. The aj are smooth functions on M, and aj(x) is determined by the metric and its derivatives at x. -
A Few Remarks on Values of Hurwitz Zeta Function at Natural and Rational
A FEW REMARKS ON VALUES OF HURWITZ ZETA FUNCTION AT NATURAL AND RATIONAL ARGUMENTS PAWELJ. SZABLOWSKI Abstract. We exploit some properties of the Hurwitz zeta function ζ(n,x) 1 ∞ n in order to study sums of the form πn Pj=−∞ 1/(jk + l) and 1 ∞ j n πn Pj=−∞(−1) /(jk + l) for 2 ≤ n,k ∈ N, and integer l ≤ k/2. We show that these sums are algebraic numbers. We also show that 1 < n ∈ N and p ∈ Q ∩ (0, 1) : the numbers (ζ(n,p) + (−1)nζ(n, 1 − p))/πn are algebraic. On the way we find polynomials sm and cm of order respectively 2m + 1 and 2m + 2 such that their n−th coefficients of sine and cosine Fourier transforms are equal to (−1)n/n2m+1 and (−1)n/n2m+2 respectively. 1. Introduction Firstly we find polynomials sm and cm of orders respectively 2m +1 and 2m +2 such that their n th coefficients in respectively sine and cosine Fourier series are of the form ( 1)n−/n2m+1 and ( 1)n/n2m+2. Secondly using these polynomials we study sums of− the form: − ∞ 1 ∞ ( 1)j S(n,k,l)= , Sˆ(n,k,l)= − . (jk + l)n (jk + l)n j=−∞ j=−∞ X X for n, k, l N such that n 2, k 2 generalizing known result of Euler (recently proved differently∈ by Elkies≥ ([7]))≥ stating that the number S(n, 4, 1)/πn is ratio- nal. In particular we show that the numbers S(n,k,j)/πn and Sˆ(n,k,j)/πn for N n, k 2, and 1 j < k are algebraic.