Chapter 2 Generating Functions
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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. -
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,.. -
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. -
A Unified Approach to Generalized Stirling Numbers
ADVANCES IN APPLIED MATHEMATICS 20, 366]384Ž. 1998 ARTICLE NO. AM980586 A Unified Approach to Generalized Stirling Numbers Leetsch C. Hsu Institute of Mathematics, Dalian Uni¨ersity of Technology, Dalian 116024, China View metadata, citation and similar papers at core.ac.ukand brought to you by CORE provided by Elsevier - Publisher Connector Peter Jau-Shyong Shiue Department of Mathematical Sciences, Uni¨ersity of Ne¨ada, Las Vegas, Las Vegas, Ne¨ada 89154-4020 Received January 1, 1995; accepted January 3, 1998 It is shown that various well-known generalizations of Stirling numbers of the first and second kinds can be unified by starting with transformations between generalized factorials involving three arbitrary parameters. Previous extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Tsylova, and others are included as particular cases of our unified treatment. We have also investigated some basic properties related to our general pattern. Q 1998 Academic Press 1. INTRODUCTION As may be observed, a natural approach to generalizing Stirling numbers is to define Stirling number pairs as connection coefficients of linear transformations between generalized factorials. Of course, any useful generalization should directly imply some interesting special cases that have certain applications. The whole approach adopted in this paper is entirely different from that of Hsu and Yuwx 13 , starting with generating functions, and various new results are provided by a unified treatment. A recent paperwx 24 of TheoretÂÃ investigated some generating functions for the solutions of a type of linear partial difference equation with first-degree polynomial coefficients. Theoretically, it also provides a way of 366 0196-8858r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. -
New Bell–Sheffer Polynomial Sets
axioms Article New Bell–Sheffer Polynomial Sets Pierpaolo Natalini 1,* and Paolo Emilio Ricci 2 1 Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo, 1, 00146 Roma, Italy 2 Sezione di Matematica, International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy; [email protected] * Correspondence: [email protected] Received: 20 July 2018; Accepted: 2 October 2018; Published: 8 October 2018 Abstract: In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear. Keywords: Sheffer polynomials; generating functions; monomiality principle; shift operators; combinatorial analysis 1. Introduction In recent articles [1,2], new sets of Sheffer [3] and Brenke [4] polynomials, based on higher order Bell numbers [2,5–7], have been studied. -
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. -
Expression and Computation of Generalized Stirling Numbers
Expression and Computation of Generalized Stirling Numbers Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL 61702-2900, USA August 2, 2012 Abstract Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and general- ized divided difference. Three algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include a comprehensive algorithm using the character- ization of Riordan arrays. AMS Subject Classification: 05A15, 65B10, 33C45, 39A70, 41A80. Key Words and Phrases: Stirling numbers of the first kind, Stir- ling numbers of the second kind, factorial polynomials, generalized factorial, divided difference, k-Gamma functions, Pochhammer sym- bol and k-Pochhammer symbol. 1 Introduction The classical Stirling numbers of the first kind and the second kind, denoted by s(n; k) and S(n; k), respectively, can be defined via a pair of inverse relations n n X k n X [z]n = s(n; k)z ; z = S(n; k)[z]k; (1.1) k=0 k=0 with the convention s(n; 0) = S(n; 0) = δn;0, the Kronecker symbol, where z 2 C, n 2 N0 = N [ f0g, and the falling factorial polynomials [z]n = 1 2 T. X. He z(z − 1) ··· (z − n + 1). js(n; k)j presents the number of permutations of n elements with k disjoint cycles while S(n; k) gives the number of ways to partition n elements into k nonempty subsets. The simplest way to compute s(n; k) is finding the coefficients of the expansion of [z]n. -
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. -
A Generalization of Stirling Numbers of the Second Kind Via a Special Multiset
1 2 Journal of Integer Sequences, Vol. 13 (2010), 3 Article 10.2.5 47 6 23 11 A Generalization of Stirling Numbers of the Second Kind via a Special Multiset Martin Griffiths School of Education (Mathematics) University of Manchester Manchester M13 9PL United Kingdom [email protected] Istv´an Mez˝o Department of Applied Mathematics and Probability Theory Faculty of Informatics University of Debrecen H-4010 Debrecen P. O. Box 12 Hungary [email protected] Abstract Stirling numbers of the second kind and Bell numbers are intimately linked through the roles they play in enumerating partitions of n-sets. In a previous article we studied a generalization of the Bell numbers that arose on analyzing partitions of a special multiset. It is only natural, therefore, next to examine the corresponding situation for Stirling numbers of the second kind. In this paper we derive generating functions, formulae and interesting properties of these numbers. 1 Introduction In [6] we studied the enumeration of partitions of a particular family of multisets. Whereas all the n elements of a finite set are distinguishable, the elements of a multiset are allowed to 1 possess multiplicities greater than 1. A multiset may therefore be regarded as a generalization of a set. Indeed, a set is a multiset in which each of the elements has multiplicity 1. The enumeration of partitions of the special multisets we considered gave rise to families of numbers that we termed generalized near-Bell numbers. In this follow-up paper we study the corresponding generalization of Stirling numbers of the second kind. -
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. -
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?). -
Bell Polynomials and Binomial Type Sequences Miloud Mihoubi
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 2450–2459 www.elsevier.com/locate/disc Bell polynomials and binomial type sequences Miloud Mihoubi U.S.T.H.B., Faculty of Mathematics, Operational Research, B.P. 32, El-Alia, 16111, Algiers, Algeria Received 3 March 2007; received in revised form 30 April 2007; accepted 10 May 2007 Available online 25 May 2007 Abstract This paper concerns the study of the Bell polynomials and the binomial type sequences. We mainly establish some relations tied to these important concepts. Furthermore, these obtained results are exploited to deduce some interesting relations concerning the Bell polynomials which enable us to obtain some new identities for the Bell polynomials. Our results are illustrated by some comprehensive examples. © 2007 Elsevier B.V. All rights reserved. Keywords: Bell polynomials; Binomial type sequences; Generating function 1. Introduction The Bell polynomials extensively studied by Bell [3] appear as a standard mathematical tool and arise in combina- torial analysis [12]. Moreover, they have been considered as important combinatorial tools [11] and applied in many different frameworks from, we can particularly quote : the evaluation of some integrals and alterning sums [6,10]; the internal relations for the orthogonal invariants of a positive compact operator [5]; the Blissard problem [12, p. 46]; the Newton sum rules for the zeros of polynomials [9]; the recurrence relations for a class of Freud-type polynomials [4] and many others subjects. This large application of the Bell polynomials gives a motivation to develop this mathematical tool.