DOCSLIB.ORG

Bernoulli Numbers and Polynomials from a More General Point of View

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bernoulli Numbers and Polynomials from a More General Point of View IT0100106 ENTE PER LE NUOVE TECNOLOGIE, L'ENERGIA E L'AMBIENTE Serie Innovazione BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW GIUSEPPE. DATTOLI ENEA - Divisione Fisica Applicata Centra Ricerche Frascati, Roma CLEMENTE CESARANO ULM University, Department of Matematics, ULM, Germany SILVERIA LORENZUTTA ENEA -P Divisione Fisica Applicata Centra Ricerche "Ezio Clementel", Bologna RT/INN/2000/11 Questo rapporto ё stato preparato da: Servizio Edizioni Scientifiche - ENEA, Centro Ricerche Frascati, C.P. 65 - 00044 Frascati, Roma, Italia I contenuti tecnico-scientifici dei rapporti tecnici dell'ENEA rispecchiano I'opinione degli autori e non necessariamente quella dell'Ente. ISSN/1120-5571 BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW Riassunto Si applica il metodo della funzione generatrice per introdurre nuove forme di numeri e polinomi di Bernoulli che vengono utilizzati per sviluppare e per calcolare somme parziali che coinvolgono polinomi a piu indici ed a piu variabili. Si sviluppano considerazioni analoghe per i polinomi ed i numeri di Eulero. BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW Abstract We apply the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are develo- ped for the Euler numbers and polynomials. Key words: numeri di Bernoulli, polinomi di Bernoulli, polinomi di Hermite, polinomi di AppEl, somme parziali INDICE 1 -INTRODUCTION p. 7 2 - FINITE SUMS AND NEW CLASSES OF BERNOULLI NUMBERS p. 9 3 - EULER POLYNOMIALS p. 11 4 - CONCLUDING REMARKS p. 12 APPENDIX p. 15 REFERENCES p. 17 BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW 1 - INTRODUCTION In a previous paper [1] we have derived partial sums involving Hermite, Laguerre and Appell polynomials in terms of generalized Bernoulli polynomials. The new and interesting possibilities offered by this class of polynomials are better illustrated by an example relevant to the derivations of a partial sum involving two-index Hermite polynomials. To this aim we remind that: N-l (1) n=0 where Bn(x) are Bernoulli polynomials defined by the generating function: TBnOO (2) or in terms of Bernoulli numbers Bn as: Hermite polynomials with two variables and one parameter can be defined by means of the operational identity [2]: (4) and the hm n(x,y|x) are defined by the double sum: min(m,n) s m-s n-s l hmn(x,y|x) = m!n! У — (5) m ' stj s!(m-s)!(n-s)! The identity (4) can be used to state that: (6) and to introduce the following two variable one parameter Bernoulli polynomials: B hBr,s(x,y|x)= J r-qBs-khq,k(x,y|x) . (7) It is easy to note that from Eq. (4) we get: ЭхЭ e У{Вг(х)В5(у)}=ьВГ)8(х,у|х). (8) This new class of Bernoulli polynomials can be used to derive the following partial sum: M-1N-1 yrws (I m=0n=0 X z r> . , X Z В (9) w — - r M + -,— — • ywj ? . у w ywj which is a consequence of Eqs. (1,6,7,8). The generating function of this last class of Bernoulli polynomials can be shown to be provided by: ux+vy+xuv о» m n uve (10) This introductory examples shows that a wealth of implications is offered by the use of generalized forms of Bernoulli polynomials. In the forthcoming sections we will develop a more systematic analysis which yields a deeper insight into the effectiveness of this type of generalizations. 2 - FINITE SUMS AND NEW CLASSES OF BERNOULLI NUMBERS In Ref. [1] we have touched on the following new class of numbers: 21 '".f n!Bn_2sBs which are recognized as an Hermite convolution of Bernoulli numbsers on themselves. The generating functions of the HBn is provided by: 15 -z.-VH-n; (12) •1)(е1 -: which suggests the following generalization t? ~ (eat-l)(ebt2-l) n=0xi: with: 1 B"(a,b" ) = — У n ab ^ s!(n-2s)! b S—\J It is also evident that the generating function: 2 3 xt+yt tn / . 2 (eat _ 1)(ebt _ 1} " —(нВп(а,Ь|х,у)) (15) can be exploited to define the polynomials: HB*n(a,b|x,y)= ip(HB;_s(a,b)Hs(X)y)) (16) as also results from Eqs. (13) and (15) which are also expressed as: 10 [n 2] n 2s s * Пn! ^ n!a - b нцВ пn(а,v Ь i х,у,yj) = — (17) ab % s!(n-2s)! S—KJ yielding: HBn(a,b|0,0)=HBn(a,b) (18) Гп /91 V,s.,n-2 X s are the Kam de where Hn(x,y) = п!1^=0 g!(n-2s)! P^ Feriet polynomials. The use of polynomials (16) is suggested by partial sums of the type: M-lN-l 1 (r + l)(r + 2)(r + 3) m=0n=0 B B w x z Nw B ( W X ~HH r+r+33 (У' I ' + ) +H r+r+33 У' I ' A further application of polynomials HBn(a,b|x,y) is relevant to the multiplication theorems. We find indeed: n 2 h 1 B (mx, 2 N m ~ VVY „• H n m в vl = -X X н m TIT nr (20) m m^ and m-lm —1, т,-1 u i ak bh HBn a,b|x + — ,y + -j (21) k=l h=l m and: p-1 q-1, к h — 1 V 1 '% 1 (22) k=0 h=C pq can be proved by exploiting the procedure outlined in Appendix. 11 3 - EULER POLYNOMIALS The examples we have provided yields an idea of the implications offered by this type of generalization. It is also evident that the considerations we have developed for Bernoulli polynomials can be extended to Euler polynomials [3]: In analogy of Ref. [1] and the results of previous sections, we introduce the following classes of Euler polynomials: 2 2 xt+yt ~ tn ХМЕ^» (24) and: m n It is easily realized that: ySE2(X) (26) s=0 s!(n-2s)! and: m n /'щЛ/'^Л E A t c l:: I1 A l А1 h 'm>nv >.yi / ~ Zu 2-Л m-s 'n-r s,rv >.y I / \ ) s r s=o r=o^ A y where En(x) are the ordinary Euler polynomials and that the following theorems hold: : Z w) 28 _ s ...... , s^ ' ( > and (see appendix): 12 m-1 / / , \\ n k HEn(mx,py) = m X (~l) l HEn^ + ->^TJJ • (29) As to the polynomials hEm^C^yl't) wa can also state that: s r Em_sn_r(x,y|t))z w (30) and: Цx k=Oh=O pq 4 - CONCLUDING REMARKS The introduction of the Hermite-Euler polynomials given by Eq. (24) offers the possibility of speculating about alternative definitions as e.g. (t < л/тг): (32) (е1+1)(ег +1) n=0n- The polynomials Е„ (х,у) are defined as: -2s(X) (33) s! n 2s ! s=o ( - > - and are shown to satisfy the following differential equation: ( 2) ( 2) |-E n (x,y) = ^TE n (x,y) (34) Эу Эх and the multiplication theorem: m-lm -1 ( f л и \Л 2 Р "]Г J (-1)^^(2)^1^^ (35) 13 The more general theorem relevant to Е„ '(mx,py) requires the introduction of a class of Euler polynomials analogous to those provided by Eq. (13) for the Bernoulli case. Before concluding this paper we want to emphasize that most of the identities holding for the ordinary Bernoulli or Euler polynomials can be extended to the generalized case. For example the identity [4]: n 1 Bn(x + 1)-Bn(x) = nx - (36) can be generalized as: HBn(x + l,y)-HBn(x,y) = nHn_1(x,y) (37) which is a consequence of: y 2 e 3x Bn(x)=HBn(x,y). (38) For analogous reasons, we find: hBm,n(x + l,y + l|T)-hBm>n(x,y + l|T)-hBm>n(x + l,y|T) (39) B +h m,n<Xy IT) = mnhm_1>n_1(x,y I T) and: HB* (a,b | x + а,у + b) -H B^(a,b | x,y) = n(n - l)(n - 2)Hn_3(x,y) (40) and ( 2) ( 2) HE n (x + l,y)+HE n (x,y) = 2Hn(x,y). (41) We can also define the further generalized form: thus getting (n>N): 14 ({}|{j)({}|{}) ^J({}) (43) with: 2: Xit ^({}) e i=i . (44) n=0 П' The results if this paper show that the combination of operational rules and the properties of ordinary and generalized polynomials offer a wealth of possibilities to introduce new familes of Euler and Bernoulli polynomials which provides a powerful tool in applications. 15 APPENDIX The multiplication theorems are easily stated by exploiting the method of the generating functions and suitable manipulations. We note indeed that: m n (h-^m,n(Px'qy|'r) = • (A.I) m=Un=O m" П' (e ~*№ -1) The r.h.s. of the above relation can be more conveniently rewritten as: (eu-l)(ev-l) pq (euP - by nothing that: up , p-l e -1 r=0 we can rearrange (A.2) as: uveuPx+vqy+Tuv P-l Я-1 } ~ - (up)m (vq)n (A.4) X P^ pq which once confronted with (A.I) yields Eq. (22). A symilar procedure can be exploited to prove the multiplication theorems relevant to the Euler's generalized forms. We note indeed: 00 tn \ 2emxt+pyt2 X ~7 (н En(mx' РУ))= 1 (A-5) n=0n! e +1 and handling the r.h.s.
Load more

Recommended publications
  • The Unilateral Z–Transform and Generating Functions
    The Unilateral z{Transform and Generating Functions Recall from \Discrete{Time Linear, Time Invariant Systems and z-Transforms" that the behaviour of a discrete{time LTI system is determined by its impulse response function h[n] and that the z{transform of h[n] is 1 k H(z) = z− h[k] k=X −∞ If the LTI system is causal, then h[n] = 0 for all n < 0 and 1 k H(z) = z− h[k] Xk=0 Definition 1 (Unilateral z{Transform) The unilateral z{transform of the discrete{time signal x[n] (whether or not x[n] = 0 for negative n's) is defined to be 1 n (z) = z− x[n] X nX=0 When there is any danger of confusing the regular z{transform with the unilateral z{transform, 1 n X(z) = z− x[n] nX= 1 − is called the bilateral z{transform. Example 2 The signal x[n] = anu[n] is zero for all n < 0. So the unilateral z{transform of x[n] is the same as the ordinary z{transform. So, as we saw in Example 7 of \Discrete{Time Linear, Time Invariant Systems and z-Transforms", 1 n n 1 (z) = X(z) = z− a = 1 X 1 z− a nX=0 − 1 provided that z− a < 1, or equivalently z > a . Since the unilateral z{transform of any x[n] is always j j j j j j equal to the bilateral z{transform of the right{sided signal x[n]u[n], the region of convergence of a unilateral z{transform is always the exterior of a circle.
    [Show full text]
  • Ordinary Generating Functions
    CHAPTER 10 Ordinary Generating Functions Introduction We’ll begin this chapter by introducing the notion of ordinary generating functions and discussing the basic techniques for manipulating them. These techniques are merely restatements and simple applications of things you learned in algebra and calculus. You must master these basic ideas before reading further. In Section 2, we apply generating functions to the solution of simple recursions. This requires no new concepts, but provides practice manipulating generating functions. In Section 3, we return to the manipulation of generating functions, introducing slightly more advanced methods than those in Section 1. If you found the material in Section 1 easy, you can skim Sections 2 and 3. If you had some difficulty with Section 1, those sections will give you additional practice developing your ability to manipulate generating functions. Section 4 is the heart of this chapter. In it we study the Rules of Sum and Product for ordinary generating functions. Suppose that we are given a combinatorial description of the construction of some structures we wish to count. These two rules often allow us to write down an equation for the generating function directly from this combinatorial description. Without such tools, we may get bogged down in lengthy algebraic manipulations. 10.1 What are Generating Functions? In this section, we introduce the idea of ordinary generating functions and look at some ways to manipulate them. This material is essential for understanding later material on generating functions. Be sure to work the exercises in this section before reading later sections! Definition 10.1 Ordinary generating function (OGF) Suppose we are given a sequence a0,a1,..
    [Show full text]
  • An Identity for Generalized Bernoulli Polynomials
    1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.11.2 47 6 23 11 An Identity for Generalized Bernoulli Polynomials Redha Chellal1 and Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria [email protected] [email protected] [email protected] Mohamed Mehbali Centre for Research Informed Teaching London South Bank University London United Kingdom [email protected] Abstract Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli poly- nomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identi- ties for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling. 1Corresponding author. 1 1 Introduction Let N and C denote, respectively, the set of positive integers and the set of complex numbers. (α) In his book, Roman [41, p. 93] defined generalized Bernoulli polynomials Bn (x) as follows: for all n ∈ N and α ∈ C, we have ∞ tn t α B(α)(x) = etx. (1) n n! et − 1 Xn=0 The Bernoulli numbers Bn, classical Bernoulli polynomials Bn(x), and generalized Bernoulli (α) numbers Bn are, respectively, defined by (1) (α) (α) Bn = Bn(0), Bn(x)= Bn (x), and Bn = Bn (0). (2) The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory, mathematical analysis, and topology.
    [Show full text]
  • The Bloch-Wigner-Ramakrishnan Polylogarithm Function
    Math. Ann. 286, 613424 (1990) Springer-Verlag 1990 The Bloch-Wigner-Ramakrishnan polylogarithm function Don Zagier Max-Planck-Insfitut fiir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany To Hans Grauert The polylogarithm function co ~n appears in many parts of mathematics and has an extensive literature [2]. It can be analytically extended to the cut plane ~\[1, ~) by defining Lira(x) inductively as x [ Li m_ l(z)z-tdz but then has a discontinuity as x crosses the cut. However, for 0 m = 2 the modified function O(x) = ~(Liz(x)) + arg(1 -- x) loglxl extends (real-) analytically to the entire complex plane except for the points x=0 and x= 1 where it is continuous but not analytic. This modified dilogarithm function, introduced by Wigner and Bloch [1], has many beautiful properties. In particular, its values at algebraic argument suffice to express in closed form the volumes of arbitrary hyperbolic 3-manifolds and the values at s= 2 of the Dedekind zeta functions of arbitrary number fields (cf. [6] and the expository article [7]). It is therefore natural to ask for similar real-analytic and single-valued modification of the higher polylogarithm functions Li,. Such a function Dm was constructed, and shown to satisfy a functional equation relating D=(x-t) and D~(x), by Ramakrishnan E3]. His construction, which involved monodromy arguments for certain nilpotent subgroups of GLm(C), is completely explicit, but he does not actually give a formula for Dm in terms of the polylogarithm. In this note we write down such a formula and give a direct proof of the one-valuedness and functional equation.
    [Show full text]
  • A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function
    mathematics Article A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function Daeyeoul Kim 1,* and Yilmaz Simsek 2 1 Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Korea 2 Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya TR-07058, Turkey; [email protected] * Correspondence: [email protected] Abstract: In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials. Keywords: Bernoulli numbers and polynomials; Euler numbers and polynomials; Apostol–Bernoulli and Apostol–Euler numbers and polynomials; Hurwitz–Lerch zeta function; Hurwitz zeta function; alternating Hurwitz zeta function; generating function; Mellin transformation MSC: 05A15; 11B68; 26C0; 11M35 Citation: Kim, D.; Simsek, Y. A New Family of Zeta Type Function 1. Introduction Involving the Hurwitz Zeta Function The families of zeta functions and special numbers and polynomials have been studied and the Alternating Hurwitz Zeta widely in many areas. They have also been used to model real-world problems.
    [Show full text]
  • Functions of Random Variables
    Names for Eg(X ) Generating Functions Topic 8 The Expected Value Functions of Random Variables 1 / 19 Names for Eg(X ) Generating Functions Outline Names for Eg(X ) Means Moments Factorial Moments Variance and Standard Deviation Generating Functions 2 / 19 Names for Eg(X ) Generating Functions Means If g(x) = x, then µX = EX is called variously the distributional mean, and the first moment. • Sn, the number of successes in n Bernoulli trials with success parameter p, has mean np. • The mean of a geometric random variable with parameter p is (1 − p)=p . • The mean of an exponential random variable with parameter β is1 /β. • A standard normal random variable has mean0. Exercise. Find the mean of a Pareto random variable. Z 1 Z 1 βαβ Z 1 αββ 1 αβ xf (x) dx = x dx = βαβ x−β dx = x1−β = ; β > 1 x β+1 −∞ α x α 1 − β α β − 1 3 / 19 Names for Eg(X ) Generating Functions Moments In analogy to a similar concept in physics, EX m is called the m-th moment. The second moment in physics is associated to the moment of inertia. • If X is a Bernoulli random variable, then X = X m. Thus, EX m = EX = p. • For a uniform random variable on [0; 1], the m-th moment is R 1 m 0 x dx = 1=(m + 1). • The third moment for Z, a standard normal random, is0. The fourth moment, 1 Z 1 z2 1 z2 1 4 4 3 EZ = p z exp − dz = −p z exp − 2π −∞ 2 2π 2 −∞ 1 Z 1 z2 +p 3z2 exp − dz = 3EZ 2 = 3 2π −∞ 2 3 z2 u(z) = z v(t) = − exp − 2 0 2 0 z2 u (z) = 3z v (t) = z exp − 2 4 / 19 Names for Eg(X ) Generating Functions Factorial Moments If g(x) = (x)k , where( x)k = x(x − 1) ··· (x − k + 1), then E(X )k is called the k-th factorial moment.
    [Show full text]
  • The Euler-Maclaurin Summation Formula and Bernoulli Polynomials R 1 Consider Two Different Ways of Applying Integration by Parts to 0 F(X)Dx
    The Euler-Maclaurin Summation Formula and Bernoulli Polynomials R 1 Consider two different ways of applying integration by parts to 0 f(x)dx. Z 1 Z 1 Z 1 1 0 0 f(x)dx = [xf(x)]0 − xf (x)dx = f(1) − xf (x)dx (1) 0 0 0 Z 1 Z 1 Z 1 1 0 f(1) + f(0) 0 f(x)dx = [(x − 1/2)f(x)]0 − (x − 1/2)f (x)dx = − (x − 1/2)f (x)dx (2) 0 0 2 0 The constant of integration was chosen to be 0 in (1), but −1/2 in (2). Which way is better? In general R 1 (2) is better. Since 0 (x − 1/2)dx = 0, that is x − 1/2 has zero average in (0, 1), it has better chance of reducing the last integral term, which is regarded as the error in estimating the given integral. We can again apply integration by parts to (2). Then the last integral will have f 00(x) multiplied by an integral of x − 1/2. We can select the constant of integration in such a way that the new polynomial multiplying f 00(x) will have zero average in the interval. If we continue this way we will get a formula which will have a higher derivative of f in the last integral multiplying a higher degree polynomial in x. If at each step the constant of integration is chosen in such a way that the new polynomial has zero average over (0, 1), we will get the Euler-Maclaurin summation formula.
    [Show full text]
  • Generating Functions
    Mathematical Database GENERATING FUNCTIONS 1. Introduction The concept of generating functions is a powerful tool for solving counting problems. Intuitively put, its general idea is as follows. In counting problems, we are often interested in counting the number of objects of ‘size n’, which we denote by an . By varying n, we get different values of an . In this way we get a sequence of real numbers a0 , a1 , a2 , … from which we can define a power series (which in some sense can be regarded as an ‘infinite- degree polynomial’) 2 Gx()= a01++ ax ax 2 +. The above Gx() is the generating function for the sequence a0 , a1 , a2 , …. In this set of notes we will look at some elementary applications of generating functions. Before formally introducing the tool, let us look at the following example. Example 1.1. (IMO 2001 HK Preliminary Selection Contest) Find the coefficient of x17 in the expansion of (1++xx5720 ) . Solution. 17 5 7 20 The only way to form an x term is to gather two x and one x . Since there are C2 =190 ways to choose two x5 from the 20 multiplicands and 18 ways to choose one x7 from the remaining 18 multiplicands, the answer is 190×= 18 3420 . To gain a preliminary insight into how generating functions is related to counting, let us describe the above problem in another way. Suppose there are 20 bags, each containing a $5 coin and a $7 coin. If we can use at most one coin from each bag, in how many different ways can we pay $17, assuming that all coins are distinguishable (i.e.
    [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]
  • 3 Formal Power Series
    MT5821 Advanced Combinatorics 3 Formal power series Generating functions are the most powerful tool available to combinatorial enu- merators. This week we are going to look at some of the things they can do. 3.1 Commutative rings with identity In studying formal power series, we need to specify what kind of coefficients we should allow. We will see that we need to be able to add, subtract and multiply coefficients; we need to have zero and one among our coefficients. Usually the integers, or the rational numbers, will work fine. But there are advantages to a more general approach. A favourite object of some group theorists, the so-called Nottingham group, is defined by power series over a finite field. A commutative ring with identity is an algebraic structure in which addition, subtraction, and multiplication are possible, and there are elements called 0 and 1, with the following familiar properties: • addition and multiplication are commutative and associative; • the distributive law holds, so we can expand brackets; • adding 0, or multiplying by 1, don’t change anything; • subtraction is the inverse of addition; • 0 6= 1. Examples incude the integers Z (this is in many ways the prototype); any field (for example, the rationals Q, real numbers R, complex numbers C, or integers modulo a prime p, Fp. Let R be a commutative ring with identity. An element u 2 R is a unit if there exists v 2 R such that uv = 1. The units form an abelian group under the operation of multiplication. Note that 0 is not a unit (why?).
    [Show full text]
  • Formal Power Series License: CC BY-NC-SA
    Formal Power Series License: CC BY-NC-SA Emma Franz April 28, 2015 1 Introduction The set S[[x]] of formal power series in x over a set S is the set of functions from the nonnegative integers to S. However, the way that we represent elements of S[[x]] will be as an infinite series, and operations in S[[x]] will be closely linked to the addition and multiplication of finite-degree polynomials. This paper will introduce a bit of the structure of sets of formal power series and then transfer over to a discussion of generating functions in combinatorics. The most familiar conceptualization of formal power series will come from taking coefficients of a power series from some sequence. Let fang = a0; a1; a2;::: be a sequence of numbers. Then 2 the formal power series associated with fang is the series A(s) = a0 + a1s + a2s + :::, where s is a formal variable. That is, we are not treating A as a function that can be evaluated for some s0. In general, A(s0) is not defined, but we will define A(0) to be a0. 2 Algebraic Structure Let R be a ring. We define R[[s]] to be the set of formal power series in s over R. Then R[[s]] is itself a ring, with the definitions of multiplication and addition following closely from how we define these operations for polynomials. 2 2 Let A(s) = a0 + a1s + a2s + ::: and B(s) = b0 + b1s + b1s + ::: be elements of R[[s]]. Then 2 the sum A(s) + B(s) is defined to be C(s) = c0 + c1s + c2s + :::, where ci = ai + bi for all i ≥ 0.
    [Show full text]
  • Euler-Maclaurin Summation Formula
    4 Euler-Maclaurin Summation Formula 4.1 Bernoulli Number & Bernoulli Polynomial 4.1.1 Definition of Bernoulli Number Bernoulli numbers Bk ()k =1,2,3, are defined as coefficients of the following equation. x Bk k x =Σ x e -1 k=0 k! 4.1.2 Expreesion of Bernoulli Numbers B B x 0 1 B2 2 B3 3 B4 4 = + x + x + x + x + (1,1) ex-1 0! 1! 2! 3! 4! ex-1 1 x x2 x3 x4 = + + + + + (1.2) x 1! 2! 3! 4! 5! Making the Cauchy product of (1.1) and (1.1) , B1 B0 B2 B1 B0 1 = B + + x + + + x2 + 0 1!1! 0!2! 2!1! 1!2! 0!3! In order to holds this for arbitrary x , B1 B0 B2 B1 B0 B =1 , + =0 , + + =0 , 0 1!1! 0!2! 2!1! 1!2! 0!3! n The coefficient of x is as follows. Bn Bn-1 Bn-2 B1 B0 + + + + + = 0 n!1! ()n -1 !2! ()n-2 !3! 1!n ! 0!()n +1 ! Multiplying both sides by ()n +1 ! , Bn()n +1 ! Bn-1()n+1 ! Bn-2()n +1 ! B1()n +1 ! B0 + + + + + = 0 n !1! ()n-1 !2! ()n -2 !3! 1!n ! 0! Using binomial coefficiets, n +1Ｃn Bn + n +1Ｃn-1 Bn-1 + n +1Ｃn-2 Bn-2 + + n +1Ｃ 1 B1 + n +1Ｃ 0 B0 = 0 Replacing n +1 with n , (1.3) nnＣ -1 Bn-1 + nnＣ -2 Bn-2 + nnＣ -3 Bn-3 + + nＣ1 B1 + nＣ0 B0 = 0 Substituting n =2, 3, 4, for this one by one, 1 21Ｃ B + 20Ｃ B = 0 B = - 1 0 2 2 1 32Ｃ B + 31Ｃ B + 30Ｃ B = 0 B = 2 1 0 2 6 Thus, we obtain Bernoulli numbers.
    [Show full text]