The Pentagonal Number Theorem and Modular Forms
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Generalized Modular Forms Representable As Eta Products
Modular Forms Representable As Eta Products A Dissertation Submitted to the Temple University Graduate Board in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY by Wissam Raji August, 2006 iii c by Wissam Raji August, 2006 All Rights Reserved iv ABSTRACT Modular Forms Representable As Eta Products Wissam Raji DOCTOR OF PHILOSOPHY Temple University, August, 2006 Professor Marvin Knopp, Chair In this dissertation, we discuss modular forms that are representable as eta products and generalized eta products . Eta products appear in many areas of mathematics in which algebra and analysis overlap. M. Newman [15, 16] published a pair of well-known papers aimed at using eta-product to construct forms on the group Γ0(n) with the trivial multiplier system. Our work here divides into three related areas. The first builds upon the work of Siegel [23] and Rademacher [19] to derive modular transformation laws for functions defined as eta products (and related products). The second continues work of Kohnen and Mason [9] that shows that, under suitable conditions, a generalized modular form is an eta product or generalized eta product and thus a classical modular form. The third part of the dissertation applies generalized eta-products to rederive some arithmetic identities of H. Farkas [5, 6]. v ACKNOWLEDGEMENTS I would like to thank all those who have helped in the completion of my thesis. To God, the beginning and the end, for all His inspiration and help in the most difficult days of my life, and for all the people listed below. To my advisor, Professor Marvin Knopp, a great teacher and inspirer whose support and guidance were crucial for the completion of my work. -
Integer Partitions
12 PETER KOROTEEV Remark. Note that out n! fixed points of T acting on Xn only for single point q we 12 PETER KOROTEEV get a polynomial out of Vq.Othercoefficient functions Vp remain infinite series which we 12shall further ignore. PETER One KOROTEEV can verify by examining (2.25) that in the n limit these Remark. Note that out n! fixed points of T acting on Xn only for single point q we coefficient functions will be suppressed. A choice of fixed point q following!1 the large-n limits get a polynomial out of Vq.Othercoefficient functions Vp remain infinite series which we Remark. Note thatcorresponds out n! fixed to zooming points of intoT aacting certain on asymptoticXn only region for single in which pointa q we a shall further ignore. One can verify by examining (2.25) that in the n | Slimit(1)| these| S(2)| ··· get a polynomial out ofaVSq(n.Othercoe) ,whereS fficientSn is functionsa permutationVp remain corresponding infinite to series the choice!1 which of we fixed point q. shall furthercoe ignore.fficient functions One| can| will verify be suppressed. by2 examining A choice (2.25 of) that fixed in point theqnfollowinglimit the large- thesen limits corresponds to zooming into a certain asymptotic region in which a!1 a coefficient functions will be suppressed. A choice of fixed point q following| theS(1) large-| | nSlimits(2)| ··· aS(n) ,whereS Sn is a permutation corresponding to the choice of fixed point q. q<latexit sha1_base64="gqhkYh7MBm9GaiVNDBxVo5M1bBY=">AAAB8HicbVBNS8NAEN3Ur1q/qh69LBbBU0lEUG9FLx4rWFtsQ9lsJ+3SzSbuTsQS+i+8eFDx6s/x5r9x2+agrQ8GHu/NMDMvSKQw6LrfTmFpeWV1rbhe2tjc2t4p7+7dmTjVHBo8lrFuBcyAFAoaKFBCK9HAokBCMxheTfzmI2gjYnWLowT8iPWVCAVnaKX7DsITBmH2MO6WK27VnYIuEi8nFZKj3i1/dXoxTyNQyCUzpu25CfoZ0yi4hHGpkxpIGB+yPrQtVSwC42fTi8f0yCo9GsbalkI6VX9PZCwyZhQFtjNiODDz3kT8z2unGJ77mVBJiqD4bFGYSooxnbxPe0IDRzmyhHEt7K2UD5hmHG1IJRuCN//yImmcVC+q3s1ppXaZp1EkB+SQHBOPnJEauSZ10iCcKPJMXsmbY5wX5935mLUWnHxmn/yB8/kDhq2RBA==</latexit> -
1 + 4 + 7 + 10 = 22;... the Nth Pentagonal Number Is Therefore
CHAPTER 6 1. The pentagonal numbers are 1; 1 + 4 = 5; 1 + 4 + 7 = 12; 1 + 4 + 7 + 10 = 22;... The nth pentagonal number is therefore n 1 2 − n(n 1) 3n n (3i +1) = n + 3 − = − . i 2 ! 2 X=0 Similarly, since the hexagonal numbers are 1; 1+5 = 6; 1+5+9 = 15; 1+5+9+13 = 28;... it follows that the nth hexagonal number is n 1 − n(n 1) (4i +1) = n + 4 − = 2n2 n. i 2 ! − X=0 2. The pyramidal numbers with triangular base are 1; 1+3 = 4; 1+3+6 = 10; 1+3+6+10 = 20; ... Therefore the nth pyramidal number with triangular base is n n k(k + 1) 1 1 n(n + 1)(2n + 1) n(n + 1) = (k2 + k)= + k 2 2 k 2 " 6 2 # X=1 X=1 . n(n + 1) 2n + 1 1 n(n + 1)(n + 2) = + = 2 6 2 6 The pyramidal numbers with square base are 1; 1+4 = 5; 1+4+9 = 14; 1+4+9+16= 30;... Thus the nth pyramidal number with square base is given by the sum of the squares from 1 to n, namely, n(n + 1)(2n + 1) . 6 3. In a harmonic proportion, c : a =(c b):(b a). It follows that ac ab = bc ac or that b(a + c) = 2ac. Thus the sum of− the extremes− multiplied by the mean− equals− twice the product of the extremes. 4. Since 6:3=(5 3) : (6 5), the numbers 3, 5, 6 are in subcontrary proportion. -
PENTAGONAL NUMBERS in the PELL SEQUENCE and DIOPHANTINE EQUATIONS 2X2 = Y2(3Y -1) 2 ± 2 Ve Siva Rama Prasad and B
PENTAGONAL NUMBERS IN THE PELL SEQUENCE AND DIOPHANTINE EQUATIONS 2x2 = y2(3y -1) 2 ± 2 Ve Siva Rama Prasad and B. Srlnivasa Rao Department of Mathematics, Osmania University, Hyderabad - 500 007 A.P., India (Submitted March 2000-Final Revision August 2000) 1. INTRODUCTION It is well known that a positive integer N is called a pentagonal (generalized pentagonal) number if N = m(3m ~ 1) 12 for some integer m > 0 (for any integer m). Ming Leo [1] has proved that 1 and 5 are the only pentagonal numbers in the Fibonacci sequence {Fn}. Later, he showed (in [2]) that 2, 1, and 7 are the only generalized pentagonal numbers in the Lucas sequence {Ln}. In [3] we have proved that 1 and 7 are the only generalized pentagonal numbers in the associated Pell sequence {Qn} delned by Q0 = Qx = 1 and Qn+2 = 2Qn+l + Q„ for n > 0. (1) In this paper, we consider the Pell sequence {PJ defined by P 0 =0,P 1 = 1, and Pn+2=2Pn+l+Pn for«>0 (2) and prove that P±l, P^, P4, and P6 are the only pentagonal numbers. Also we show that P0, P±1, P2, P^, P4, and P6 are the only generalized pentagonal numbers. Further, we use this to solve the Diophantine equations of the title. 2. PRELIMINARY RESULTS We have the following well-known properties of {Pn} and {Qn}: for all integers m and n, a a + pn = "-P" a n d Q = " P" w herea = 1 + V2 and p = 1-V2, (3) l P_„ = {-\r Pn and Q_„ = (-l)"Qn, (4) a2 = 2P„2+(-iy, (5) 2 2 e3„=a(e„ +6P„ ), (6> ^ + n = 2PmG„-(-l)"Pm_„. -
Modular Dedekind Symbols Associated to Fuchsian Groups and Higher-Order Eisenstein Series
Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series Jay Jorgenson,∗ Cormac O’Sullivan†and Lejla Smajlovi´c May 20, 2018 Abstract Let E(z,s) be the non-holomorphic Eisenstein series for the modular group SL(2, Z). The classical Kro- necker limit formula shows that the second term in the Laurent expansion at s = 1 of E(z,s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressedits multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups Γ. The analogue of the eta function has a multiplier system that may be expressed in terms of a map S : Γ → R which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from Γ to C produces higher-order Eisenstein series. These series share many of the properties of E(z,s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher- order Dedekind symbols S∗ :Γ → R. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups Γ0(N) are rational. 1 Introduction 1.1 Kronecker limit functions and Dedekind sums Write elements of the upper half plane H as z = x + iy with y > 0. The non-holomorphic Eisenstein series E (s,z), associated to the full modular group SL(2, Z), may be given as ∞ 1 ys E (z,s) := . -
Torus N-Point Functions for $\Mathbb {R} $-Graded Vertex Operator
Torus n-Point Functions for R-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds Geoffrey Mason∗ Department of Mathematics, University of California Santa Cruz, CA 95064, U.S.A. Michael P. Tuite, Alexander Zuevsky† Department of Mathematical Physics, National University of Ireland, Galway, Ireland. October 23, 2018 Abstract We consider genus one n-point functions for a vertex operator su- peralgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the arXiv:0708.0640v1 [math.QA] 4 Aug 2007 rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orb- ifold n-point functions are given and we describe a generalization of Fay’s trisecant identity for elliptic functions. ∗Partial support provided by NSF, NSA and the Committee on Research, University of California, Santa Cruz †Supported by a Science Foundation Ireland Frontiers of Research Grant, and by Max- Planck Institut f¨ur Mathematik, Bonn 1 1 Introduction This paper is one of a series devoted to the study of n-point functions for vertex operator algebras on Riemann surfaces of genus one, two and higher [T], [MT1], [MT2], [MT3]. One may define n-point functions at genus one following Zhu [Z], and use these functions together with various sewing pro- cedures to define n-point functions at successively higher genera [T], [MT2], [MT3]. In this paper we consider the genus one n-point functions for a Vertex Operator Superalgebra (VOSA) V with a real grading (i.e. -
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) .. -
Euler's Pentagonal Number Theorem
Euler's Pentagonal Number Theorem Dan Cranston September 28, 2011 Triangular Numbers:1 ; 3; 6; 10; 15; 21; 28; 36; 45; 55; ::: Square Numbers:1 ; 4; 9; 16; 25; 36; 49; 64; 81; 100; ::: Pentagonal Numbers:1 ; 5; 12; 22; 35; 51; 70; 92; 117; 145; ::: Introduction Square Numbers:1 ; 4; 9; 16; 25; 36; 49; 64; 81; 100; ::: Pentagonal Numbers:1 ; 5; 12; 22; 35; 51; 70; 92; 117; 145; ::: Introduction Triangular Numbers:1 ; 3; 6; 10; 15; 21; 28; 36; 45; 55; ::: Pentagonal Numbers:1 ; 5; 12; 22; 35; 51; 70; 92; 117; 145; ::: Introduction Triangular Numbers:1 ; 3; 6; 10; 15; 21; 28; 36; 45; 55; ::: Square Numbers:1 ; 4; 9; 16; 25; 36; 49; 64; 81; 100; ::: Introduction Triangular Numbers:1 ; 3; 6; 10; 15; 21; 28; 36; 45; 55; ::: Square Numbers:1 ; 4; 9; 16; 25; 36; 49; 64; 81; 100; ::: Pentagonal Numbers:1 ; 5; 12; 22; 35; 51; 70; 92; 117; 145; ::: The kth pentagonal number, P(k), is the kth partial sum of the arithmetic sequence an = 1 + 3(n − 1) = 3n − 2. k X 3k2 − k P(k) = (3n − 2) = 2 n=1 I P(8) = 92, P(500) = 374; 750, etc. and P(0) = 0. I Extend domain, so P(−8) = 100, P(−500) = 375; 250, etc. I fP(0); P(1); P(−1); P(2); P(−2); :::g = f0; 1; 2; 5; 7; :::g is an increasing sequence. Generalized Pentagonal Numbers k X 3k2 − k P(k) = (3n − 2) = 2 n=1 I P(8) = 92, P(500) = 374; 750, etc. -
Modular Forms in String Theory and Moonshine
(Mock) Modular Forms in String Theory and Moonshine Miranda C. N. Cheng Korteweg-de-Vries Institute of Mathematics and Institute of Physics, University of Amsterdam, Amsterdam, the Netherlands Abstract Lecture notes for the Asian Winter School at OIST, Jan 2016. Contents 1 Lecture 1: 2d CFT and Modular Objects 1 1.1 Partition Function of a 2d CFT . .1 1.2 Modular Forms . .4 1.3 Example: The Free Boson . .8 1.4 Example: Ising Model . 10 1.5 N = 2 SCA, Elliptic Genus, and Jacobi Forms . 11 1.6 Symmetries and Twined Functions . 16 1.7 Orbifolding . 18 2 Lecture 2: Moonshine and Physics 22 2.1 Monstrous Moonshine . 22 2.2 M24 Moonshine . 24 2.3 Mock Modular Forms . 27 2.4 Umbral Moonshine . 31 2.5 Moonshine and String Theory . 33 2.6 Other Moonshine . 35 References 35 1 1 Lecture 1: 2d CFT and Modular Objects We assume basic knowledge of 2d CFTs. 1.1 Partition Function of a 2d CFT For the convenience of discussion we focus on theories with a Lagrangian description and in particular have a description as sigma models. This in- cludes, for instance, non-linear sigma models on Calabi{Yau manifolds and WZW models. The general lessons we draw are however applicable to generic 2d CFTs. An important restriction though is that the CFT has a discrete spectrum. What are we quantising? Hence, the Hilbert space V is obtained by quantising LM = the free loop space of maps S1 ! M. Recall that in the usual radial quantisation of 2d CFTs we consider a plane with 2 special points: the point of origin and that of infinity. -
Generalized Jacobi Theta Functions, Macdonald's
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarBank@NUS GENERALIZED JACOBI THETA FUNCTIONS, MACDONALD’S IDENTITIES AND POWERS OF DEDEKIND’S ETA FUNCTION TOH PEE CHOON (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements I would like to thank my thesis advisor Professor Chan Heng Huat, for his invalu- able guidance throughout these four years. Without his dedication and his brilliant insights, I would not have been able to complete this thesis. I would also like to thank Professor Shaun Cooper, whom I regard as my unofficial advisor. He taught me everything that I know about the Macdonald identities, and helped to improve and refine most of the work that I have done for this thesis. Lastly, I would like to thank Professor Liu Zhi-Guo because I first learnt about the Jacobi theta functions from his excellent research papers. Toh Pee Choon 2007 ii Contents Acknowledgements ii Summary v List of Tables vii 1 Jacobi theta functions 1 1.1 Classical Jacobi theta functions . 2 1.2 Generalized Jacobi theta functions . 6 2 Powers of Dedekind’s eta function 9 2.1 Theorems of Ramanujan, Newman and Serre . 9 2.2 The eighth power of η(τ)........................ 13 2.3 The tenth power of η(τ) ........................ 20 2.4 The fourteenth power of η(τ) ..................... 23 2.5 Modular form identities . 25 iii Contents iv 2.6 The twenty-sixth power of η(τ) ................... -
On a Lattice Generalisation of the Logarithm and a Deformation of the Dedekind Eta Function
Bull. Aust. Math. Soc. 102 (2020), 118–125 doi:10.1017/S000497272000012X ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION LAURENT BETERMIN´ (Received 14 September 2019; accepted 14 January 2020; first published online 20 February 2020) Abstract (m) We consider a deformation EL;Λ(it) of the Dedekind eta function depending on two d-dimensional simple lattices (L; Λ) and two parameters (m; t) 2 (0; 1), initially proposed by Terry Gannon. We show that the (m) minimisers of the lattice theta function are the maximisers of EL;Λ(it) in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms. 2010 Mathematics subject classification: primary 33E20; secondary 11F20, 49K30. Keywords and phrases: Dedekind eta function, theta function, lattice, optimisation. 1. Introduction and setting Many mathematical models from physics are written in terms of special functions whose properties give fundamental information about the system (see, for example, [16]). For example, properties of the Jacobi theta function and the Dedekind eta function defined for =(τ) > 0 by X −iπk2τ 1=24 Y n 2iπτ θ3(τ):= e and η(τ):= q (1 − q ); q = e ; (1.1) k2Z n2N have been widely used to identify ground states of periodic systems (see, for example, [4, 10, 13, 19]). Generalisations and deformations of these special functions which arise in more complex physical systems are also of great interest. -
Figurate Numbers: Presentation of a Book
Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Figurate Numbers: presentation of a book Elena DEZA and Michel DEZA Moscow State Pegagogical University, and Ecole Normale Superieure, Paris October 2011, Fields Institute Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview 1 Overview 2 Chapter 1. Plane figurate numbers 3 Chapter 2. Space figurate numbers 4 Chapter 3. Multidimensional figurate numbers 5 Chapter 4. Areas of Number Theory including figurate numbers 6 Chapter 5. Fermat’s polygonal number theorem 7 Chapter 6. Zoo of figurate-related numbers 8 Chapter 7. Exercises 9 Index Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter 0. Overview Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview Figurate numbers, as well as a majority of classes of special numbers, have long and rich history. They were introduced in Pythagorean school (VI -th century BC) as an attempt to connect Geometry and Arithmetic. Pythagoreans, following their credo ”all is number“, considered any positive integer as a set of points on the plane. Overview Chapter 1. Plane figurate numbers Chapter 2. Space figurate numbers Chapter 3. Multidimensional figurate numbers Chapter Overview In general, a figurate number is a number that can be represented by regular and discrete geometric pattern of equally spaced points. It may be, say, a polygonal, polyhedral or polytopic number if the arrangement form a regular polygon, a regular polyhedron or a reqular polytope, respectively.