DOCSLIB.ORG

Summability Calculus

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Summability Calculus 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.
Load more

Recommended publications
  • Accessing Bernoulli-Numbers by Matrix-Operations Gottfried Helms 3'2006 Version 2.3
    Accessing Bernoulli-Numbers by Matrix-Operations Gottfried Helms 3'2006 Version 2.3 Accessing Bernoulli-Numbers by Matrixoperations ........................................................................ 2 1. Introduction....................................................................................................................................................................... 2 2. A common equation of recursion (containing a significant error)..................................................................................... 3 3. Two versions B m and B p of bernoulli-numbers? ............................................................................................................... 4 4. Computation of bernoulli-numbers by matrixinversion of (P-I) ....................................................................................... 6 5. J contains the eigenvalues, and G m resp. G p contain the eigenvectors of P z resp. P s ......................................................... 7 6. The Binomial-Matrix and the Matrixexponential.............................................................................................................. 8 7. Bernoulli-vectors and the Matrixexponential.................................................................................................................... 8 8. The structure of the remaining coefficients in the matrices G m - and G p........................................................................... 9 9. The original problem of Jacob Bernoulli: "Powersums" - from G
    [Show full text]
  • 0.999… = 1 an Infinitesimal Explanation Bryan Dawson
    0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder .
    [Show full text]
  • Understanding Student Use of Differentials in Physics Integration Problems
    PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 9, 020108 (2013) Understanding student use of differentials in physics integration problems Dehui Hu and N. Sanjay Rebello* Department of Physics, 116 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506-2601, USA (Received 22 January 2013; published 26 July 2013) This study focuses on students’ use of the mathematical concept of differentials in physics problem solving. For instance, in electrostatics, students need to set up an integral to find the electric field due to a charged bar, an activity that involves the application of mathematical differentials (e.g., dr, dq). In this paper we aim to explore students’ reasoning about the differential concept in physics problems. We conducted group teaching or learning interviews with 13 engineering students enrolled in a second- semester calculus-based physics course. We amalgamated two frameworks—the resources framework and the conceptual metaphor framework—to analyze students’ reasoning about differential concept. Categorizing the mathematical resources involved in students’ mathematical thinking in physics provides us deeper insights into how students use mathematics in physics. Identifying the conceptual metaphors in students’ discourse illustrates the role of concrete experiential notions in students’ construction of mathematical reasoning. These two frameworks serve different purposes, and we illustrate how they can be pieced together to provide a better understanding of students’ mathematical thinking in physics. DOI: 10.1103/PhysRevSTPER.9.020108 PACS numbers: 01.40.Àd mathematical symbols [3]. For instance, mathematicians I. INTRODUCTION often use x; y asR variables and the integrals are often written Mathematical integration is widely used in many intro- in the form fðxÞdx, whereas in physics the variables ductory and upper-division physics courses.
    [Show full text]
  • Leonhard Euler: His Life, the Man, and His Works∗
    SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St.
    [Show full text]
  • Almost Periodic Sequences and Functions with Given Values
    ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 1–16 ALMOST PERIODIC SEQUENCES AND FUNCTIONS WITH GIVEN VALUES Michal Veselý Abstract. We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set X in a metric space, it is proved the existence of an almost periodic sequence {ψk}k∈Z such that {ψk; k ∈ Z} = X and ψk = ψk+lq(k), l ∈ Z for all k and some q(k) ∈ N which depends on k. 1. Introduction The aim of this paper is to construct almost periodic sequences and functions which attain values in a metric space. More precisely, our aim is to find almost periodic sequences and functions whose ranges contain or consist of arbitrarily given subsets of the metric space satisfying certain conditions. We are motivated by the paper [3] where a similar problem is investigated for real-valued sequences. In that paper, using an explicit construction, it is shown that, for any bounded countable set of real numbers, there exists an almost periodic sequence whose range is this set and which attains each value in this set periodically. We will extend this result to sequences attaining values in any metric space. Concerning almost periodic sequences with indices k ∈ N (or asymptotically almost periodic sequences), we refer to [4] where it is proved that, for any precompact sequence {xk}k∈N in a metric space X , there exists a permutation P of the set of positive integers such that the sequence {xP (k)}k∈N is almost periodic.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Infinitesimal Calculus
    Infinitesimal Calculus Δy ΔxΔy and “cannot stand” Δx • Derivative of the sum/difference of two functions (x + Δx) ± (y + Δy) = (x + y) + Δx + Δy ∴ we have a change of Δx + Δy. • Derivative of the product of two functions (x + Δx)(y + Δy) = xy + Δxy + xΔy + ΔxΔy ∴ we have a change of Δxy + xΔy. • Derivative of the product of three functions (x + Δx)(y + Δy)(z + Δz) = xyz + Δxyz + xΔyz + xyΔz + xΔyΔz + ΔxΔyz + xΔyΔz + ΔxΔyΔ ∴ we have a change of Δxyz + xΔyz + xyΔz. • Derivative of the quotient of three functions x Let u = . Then by the product rule above, yu = x yields y uΔy + yΔu = Δx. Substituting for u its value, we have xΔy Δxy − xΔy + yΔu = Δx. Finding the value of Δu , we have y y2 • Derivative of a power function (and the “chain rule”) Let y = x m . ∴ y = x ⋅ x ⋅ x ⋅...⋅ x (m times). By a generalization of the product rule, Δy = (xm−1Δx)(x m−1Δx)(x m−1Δx)...⋅ (xm −1Δx) m times. ∴ we have Δy = mx m−1Δx. • Derivative of the logarithmic function Let y = xn , n being constant. Then log y = nlog x. Differentiating y = xn , we have dy dy dy y y dy = nxn−1dx, or n = = = , since xn−1 = . Again, whatever n−1 y dx x dx dx x x x the differentials of log x and log y are, we have d(log y) = n ⋅ d(log x), or d(log y) n = . Placing these values of n equal to each other, we obtain d(log x) dy d(log y) y dy = .
    [Show full text]
  • 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
    [Show full text]
  • The Structure of Bernoulli Numbers
    The structure of Bernoulli numbers Bernd C. Kellner Abstract We conjecture that the structure of Bernoulli numbers can be explicitly given in the closed form n 2 −1 −1 n−l −1 −1 Bn = (−1) |n|p |p (χ(p,l) − p−1 )|p p p,l ∈ irr p−1∤n ( ) Ψ1 p−1|n Y n≡l (modY p−1) Y irr where the χ(p,l) are zeros of certain p-adic zeta functions and Ψ1 is the set of irregular pairs. The more complicated but improbable case where the conjecture does not hold is also handled; we obtain an unconditional structural formula for Bernoulli numbers. Finally, applications are given which are related to classical results. Keywords: Bernoulli number, Kummer congruences, irregular prime, irregular pair of higher order, Riemann zeta function, p-adic zeta function Mathematics Subject Classification 2000: 11B68 1 Introduction The classical Bernoulli numbers Bn are defined by the power series ∞ z zn = B , |z| < 2π , ez − 1 n n! n=0 X where all numbers Bn are zero with odd index n > 1. The even-indexed rational 1 1 numbers Bn alternate in sign. First values are given by B0 = 1, B1 = − 2 , B2 = 6 , 1 arXiv:math/0411498v1 [math.NT] 22 Nov 2004 B4 = − 30 . Although the first numbers are small with |Bn| < 1 for n = 2, 4,..., 12, these numbers grow very rapidly with |Bn|→∞ for even n →∞. For now, let n be an even positive integer. An elementary property of Bernoulli numbers is the following discovered independently by T. Clausen [Cla40] and K.
    [Show full text]
  • Tight Bounds on the Mutual Coherence of Sensing Matrices for Wigner D-Functions on Regular Grids
    Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids Arya Bangun, Arash Behboodi, and Rudolf Mathar,∗ October 7, 2020 Abstract Many practical sampling patterns for function approximation on the rotation group utilizes regu- lar samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which ap- pear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics. 1 Introduction In many applications, the goal is to recover a function defined on a group, say on the sphere S2 and the rotation group SO(3), from only a few samples [1{5].
    [Show full text]
  • 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
    [Show full text]
  • Approximation Properties of G-Bernoulli Polynomials
    Approximation properties of q-Bernoulli polynomials M. Momenzadeh and I. Y. Kakangi Near East University Lefkosa, TRNC, Mersiin 10, Turkey Email: [email protected] [email protected] July 10, 2018 Abstract We study the q−analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials is discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches, first solving differential equations by estimating functions and second we may apply these techniques for operator theory. 1 Introduction The present study has sough to investigate the approximation of suitable function f(x) as a linear com- bination of q−Bernoulli polynomials. This study by using q−operators through a parallel way has achived to the kind of Euler expansion for f(x), and expanded the function in terms of q−Bernoulli polynomials. This expansion offers a proper tool to solve q-difference equations or normal differential equation as well. There are many approaches to approximate the capable functions. According to properties of q-functions many q-function have been used in order to approximate a suitable function.For example, at [28], some identities and formulae for the q-Bernstein basis function, including the partition of unity property, formulae for representing the monomials were studied. In addition, a kind of approximation of a function in terms of Bernoulli polynomials are used in several approaches for solving differential equations, such as [7, 20, 21].
    [Show full text]