Summability Calculus Ibrahim M. Alabdulmohsin [email protected] Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) Division King Abdullah University of Science and Technology (KAUST) Thuwal 23955-6900, Kingdom of Saudi Arabia September 27, 2012 arXiv:1209.5739v1 [math.CA] 25 Sep 2012 Abstract In this paper, we present the foundations of Summability Calculus, which places var- ious established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite differences under a single simple umbrella. Using Summability Calculus, any given finite sum of the form Pn f(n) = k=a sk g(k; n), where sk is an arbitrary periodic sequence, becomes immediately in analytic form. Not only can we differentiate and integrate with respect to the bound n without having to rely on an explicit analytic formula for the finite sum, but we can also deduce asymptotic expansions, accelerate convergence, assign natural values to divergent sums, and evaluate the finite sum for any n 2 C. This follows because the Pn discrete definition of the simple finite sum f(n) = k=a sk g(k; n) embodies a unique natural definition for all n 2 C. Throughout the paper, many established results are strengthened such as the Bohr-Mollerup theorem, Stirling's approximation, Glaisher's approximation, and the Shannon-Nyquist sampling theorem. In addition, many celebrated theorems are extended and generalized such as the Euler-Maclaurin summation formula and Boole's summation formula. Finally, we show that countless identities that have been proved throughout the past 300 years by different mathematicians using different approaches can actually be derived in an elementary straightforward manner using the rules of Summability Calculus. Contents 1 Introduction 4 1.1 Preliminary Discussion . .4 1.2 Terminology and Notation . 10 1.3 Historical Remarks . 11 1.4 Outline of Work . 14 2 Simple Finite Sums 15 2.1 Foundations . 15 2.2 Examples to Foundational Rules . 21 2.3 Semi-Linear Simple Finite Sums . 23 2.4 Examples to Semi-Linear Simple Finite Sums . 30 2.4.1 Example I: Telescoping Sums . 30 2.4.2 Example II: The Factorial Function . 31 2.4.3 Example III: Euler's Constant . 33 2.4.4 Example IV: More into the Zeta Function . 35 2.4.5 Example V: One More Function . 36 2.4.6 Example VI: Asymptotic Expressions . 37 2.5 Simple Finite Sums: The General Case . 38 2.6 Examples to the General Case of Simple Finite Sums . 45 2.6.1 Example I: Extending the Bernoulli-Faulhaber formula . 45 2.6.2 Example II: The Hyperfactorial Function . 47 2.6.3 Example III: The Superfactorial Function . 47 2.6.4 Example IV: Alternating Sums . 49 2.7 Summary of Results . 50 3 Convoluted Finite Sums 51 3.1 Infinitesimal Calculus on Convoluted Sums . 51 3.2 Examples to Convoluted Finite Sums . 56 3.2.1 Example I: The Factorial Function Revisited . 56 3.2.2 Example II: Convoluted Zeta Function . 56 1 CONTENTS 2 3.2.3 Example III: Numerical Integration . 57 3.2.4 Exmaple IV: Alternating Euler-Maclaurin Sum . 58 3.2.5 Exmaple V: An identity of Ramanujan . 58 3.2.6 Exmaple VI: Limiting Behavior . 59 3.3 Summary of Results . 59 4 Analytic Summability Theory 61 4.1 The T Definition of Infinite Sums . 62 4.2 The Summability Method Ξ . 70 4.2.1 A Statement of Summability . 70 4.2.2 Convergence . 74 4.2.3 Analysis of the Error Term . 80 4.3 Summary of Results . 86 5 Oscillating Finite Sums 87 5.1 Alternating Sums . 87 5.2 Oscillating Sums: The General Case . 96 5.3 Infinitesimal Calculus on Oscillating Sums . 100 5.4 Summary of Results . 103 6 Direct Evaluation of Finite Sums 104 6.1 Evaluating Semi-Linear Simple Finite Sums . 105 6.2 Evaluating Arbitrary Simple Finite Sums . 108 6.3 Evaluating Oscillating Simple Finite Sums . 111 6.4 Evaluating Convoluted Finite Sums . 114 6.5 Summary of Results . 115 7 Summability Calculus and Finite Differences 117 7.1 Summability Calculus and the Calculus of Finite Differences . 118 7.2 Discrete Analog of the Euler-like Summation Formulas . 124 7.3 Applications to Summability Calculus . 126 7.3.1 Example I: Binomial Coefficients and Infinite Products . 126 7.3.2 Example II: The Zeta Function Revisited . 127 7.3.3 Example III: Identities Involving Gregory's Coefficients . 128 7.4 Summary of Results . 129 List of Figures 2.1 The successive polynomial approximation method . 19 2.2 Illustration of nearly convergent functions . 24 2.3 Interpreting the divergent Euler-Maclaurin summation formula . 44 p Pn 2.4 The function k=1 k plotted in the interval −1 ≤ n ≤ 1.......... 46 4.1 The summability method Ξ applied to the logarithmic function . 77 4.2 The summability method Ξ applied to the sum of square roots function . 79 4.3 A depiction of the recursive proof of the summability method Ξ . 82 p Pn 6.1 The function k=1 k plotted for n ≥ −1................... 108 n 1 1 P n 6.2 The function fG(n) = n k=1 k plotted for n > 1.............. 115 7.1 Newton's interpolation formula applied to the discretized function sin x .. 124 3 Chapter 1 Introduction One should always generalize Carl Jacobi (1804 { 1851) 1.1 Preliminary Discussion Generalization has been an often-pursued goal since the early dawn of mathematics. It can be loosely defined as the process of introducing new systems in order to extend consistently the domains of existing operations while still preserving as many prior results as possi- ble. Generalization has manifested in many areas of mathematics including fundamental concepts such as numbers and geometry, systems of operations such as the arithmetic, and even domains of functions as in analytic continuation. One historical example of mathematical generalization that is of particular interest in this paper is extending the domains of special discrete functions such as finite sums and products to non-integer arguments. Such process of generalization is quite different from mere interpolation, where the former is meant to preserve some fundamental properties of discrete functions as opposed to mere blind fitting. Consequently, generalization has intrinsic significance that provides deep insights and leads naturally to an evolution of mathematical thought. For instance if one considers the discrete power sum function Sm(n) given in Eq 1.1.1 below, it is trivial to realize that an infinite number of analytic functions can correctly interpolate it. In fact, let S be one such function, then the sum of S with any function p(n) that satisfies p(n) = 0 for all n 2 N will also interpolate correctly the discrete values of Sm(n). However, the well-known Bernoulli-Faulhaber formula for the power sum function additionally preserves the recursive property of Sm(n) given in Eq 1.1.2 for all real values of n, which makes it a suitable candidate for a generalized definition of power sums. In fact, it is indeed the unique family of polynomials that enjoys such advantage; hence it is the unique most natural generalized definition of power sums if one considers polynomials to be the simplest of all possible functions. The Bernoulli-Faulhaber formula is given in 4 CHAPTER 1. INTRODUCTION 5 1 Eq 1.1.3, where Br are Bernoulli numbers and B1 = − 2 . n X m Sm(n) = k (1.1.1) k=1 m Sm(n) = n + Sm(n − 1) (1.1.2) n 1 X m + 1 S (n) = (−1)j B nm+1−j (1.1.3) m m + 1 j j j=0 Looking into the Bernoulli-Faulhaber formula for power sums, it is not immediately obvious, without aid of the finite difference method, why Sm(n) can be a polynomial with degree m + 1, even though this fact becomes literally trivial using the simple rules of Summability Calculus presented in this paper. Of course, once a correct guess of a closed-form formula to a finite sum or product is available, it is usually straightforward to prove correctness using the method of induction. However, arriving at the right guess itself often relies on intricate ad hoc approaches. A second well-known illustrative example to the subject of mathematical generalization is extending the definition of the discrete factorial function to non-integer arguments, which had withstood many unsuccessful attempts by reputable mathematicians such as Bernoulli and Stirling until Euler came up with his famous answer in the early 1730s in a series of letters to Goldbach that introduced his infinite product formula and the Gamma function [16, 14, 60]. Indeed, arriving at the Gamma function from the discrete definition of factorial was not a simple task, needless to mention proving that it was the unique natural generalization of factorials as the Bohr-Miller theorem nearly stated [29]. Clearly, a systematic approach is needed in answering such questions so that it can be readily applied to the potentially infinite list of special discrete functions such as the factorial-like hyperfactorial and superfactorial functions, defined in Eq 1.1.4 and Eq 1.1.5 respectively (for a brief introduction to such family of functions, the reader is kindly referred to [62] and [58]). Summability Calculus provides us with the answer. n Y Hyperfactorial: H(n) = kk (1.1.4) k=1 n Y Superfactorial: S(n) = k! (1.1.5) k=1 Aside from obtaining exact closed-form generalizations to discrete functions as well as performing infinitesimal calculus, which aids in computing nth-order approximation among other applications, deducing asymptotic behavior is a third fundamental problem that could have enormous applications, too.