An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions
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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. -
Interview of Albert Tucker
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange About Harlan D. Mills Science Alliance 9-1975 Interview of Albert Tucker Terry Speed Evar Nering Follow this and additional works at: https://trace.tennessee.edu/utk_harlanabout Part of the Mathematics Commons Recommended Citation Speed, Terry and Nering, Evar, "Interview of Albert Tucker" (1975). About Harlan D. Mills. https://trace.tennessee.edu/utk_harlanabout/13 This Report is brought to you for free and open access by the Science Alliance at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in About Harlan D. Mills by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. The Princeton Mathematics Community in the 1930s (PMC39)The Princeton Mathematics Community in the 1930s Transcript Number 39 (PMC39) © The Trustees of Princeton University, 1985 ALBERT TUCKER CAREER, PART 2 This is a continuation of the account of the career of Albert Tucker that was begun in the interview conducted by Terry Speed in September 1975. This recording was made in March 1977 by Evar Nering at his apartment in Scottsdale, Arizona. Tucker: I have recently received the tapes that Speed made and find that these tapes carried my history up to approximately 1938. So the plan is to continue the history. In the late '30s I was working in combinatorial topology with not a great deal of results to show. I guess I was really more interested in my teaching. I had an opportunity to teach an undergraduate course in topology, combinatorial topology that is, classification of 2-dimensional surfaces and that sort of thing. -
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. -
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. -
E. T. Bell and Mathematics at Caltech Between the Wars Judith R
E. T. Bell and Mathematics at Caltech between the Wars Judith R. Goodstein and Donald Babbitt E. T. (Eric Temple) Bell, num- who was in the act of transforming what had been ber theorist, science fiction a modest technical school into one of the coun- novelist (as John Taine), and try’s foremost scientific institutes. It had started a man of strong opinions out in 1891 as Throop University (later, Throop about many things, joined Polytechnic Institute), named for its founder, phi- the faculty of the California lanthropist Amos G. Throop. At the end of World Institute of Technology in War I, Throop underwent a radical transformation, the fall of 1926 as a pro- and by 1921 it had a new name, a handsome en- fessor of mathematics. At dowment, and, under Millikan, a new educational forty-three, “he [had] be- philosophy [Goo 91]. come a very hot commodity Catalog descriptions of Caltech’s program of in mathematics” [Rei 01], advanced study and research in pure mathematics having spent fourteen years in the 1920s were intended to interest “students on the faculty of the Univer- specializing in mathematics…to devote some sity of Washington, along of their attention to the modern applications of with prestigious teaching mathematics” and promised “to provide defi- stints at Harvard and the nitely for such a liaison between pure and applied All photos courtesy of the Archives, California Institute of Technology. of Institute California Archives, the of courtesy photos All University of Chicago. Two Eric Temple Bell (1883–1960), ca. mathematics by the addition of instructors whose years before his arrival in 1951. -
FOCUS 8 3.Pdf
Volume 8, Number 3 THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA May-June 1988 More Voters and More Votes for MAA Mathematics and the Public Officers Under Approval Voting Leonard Gilman Over 4000 MAA mebers participated in the 1987 Association-wide election, up from about 3000 voters in the elections two years ago. Is it possible to enhance the public's appreciation of mathematics and Lida K. Barrett, was elected President Elect and Alan C. Tucker was mathematicians? Science writers are helping us by turning out some voted in as First Vice-President; Warren Page was elected Second first-rate news stories; but what about mathematics that is not in the Vice-President. Lida Barrett will become President, succeeding Leon news? Can we get across an idea here, a fact there, and a tiny notion ard Gillman, at the end of the MAA annual meeting in January of 1989. of what mathematics is about and what it is for? Is it reasonable even See the article below for an analysis of the working of approval voting to try? Yes. Herewith are some thoughts on how to proceed. in this election. LOUSY SALESPEOPLE Let us all become ambassadors of helpful information and good will. The effort will be unglamorous (but MAA Elections Produce inexpensive) and results will come slowly. So far, we are lousy salespeople. Most of us teach and think we are pretty good at it. But Decisive Winners when confronted by nonmathematicians, we abandon our profession and become aloof. We brush off questions, rejecting the opportunity Steven J. -
Degenerate Stirling Polynomials of the Second Kind and Some Applications
S S symmetry Article Degenerate Stirling Polynomials of the Second Kind and Some Applications Taekyun Kim 1,*, Dae San Kim 2,*, Han Young Kim 1 and Jongkyum Kwon 3,* 1 Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea 2 Department of Mathematics, Sogang University, Seoul 121-742, Korea 3 Department of Mathematics Education and ERI, Gyeongsang National University, Jinju 52828, Korea * Correspondence: [email protected] (T.K.); [email protected] (D.S.K.); [email protected] (J.K.) Received: 20 July 2019; Accepted: 13 August 2019; Published: 14 August 2019 Abstract: Recently, the degenerate l-Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate l-Stirling polynomials as well as the r-truncated degenerate l-Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate l-Stirling polynomials of the second and the r-truncated degenerate l-Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. Keywords: degenerate l-Stirling polynomials of the second kind; r-truncated degenerate l-Stirling polynomials of the second kind; probability distribution 1. Introduction For n ≥ 0, the Stirling numbers of the second kind are defined as (see [1–26]) n n x = ∑ S2(n, k)(x)k, (1) k=0 where (x)0 = 1, (x)n = x(x − 1) ··· (x − n + 1), (n ≥ 1). -
Sets of Iterated Partitions and the Bell Iterated Exponential Integers
Sets of iterated Partitions and the Bell iterated Exponential Integers Ivar Henning Skau University of South-Eastern Norway 3800 Bø, Telemark [email protected] Kai Forsberg Kristensen University of South-Eastern Norway 3918 Porsgrunn, Telemark [email protected] March 21, 2019 Abstract It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain partitioning process that gives rise to a sequence of sets of ”nested” partitions. We prove that at stage m, the cardinality of the resulting set will equal the m-th order Bell number. This set- theoretic interpretation enables us to make a natural definition of higher order Stirling numbers and to study the combinatorics of these entities. The cardinality of the elements of the constructed ”hyper partition” sets are explored. 1 A partitioning process. arXiv:1903.08379v1 [math.CO] 20 Mar 2019 (1) Consider the 3-set S = {a,b,c}. The partition set ℘3 of S, where the elements of S are put into boxes, contains the five partitions shown in the second column of Figure 1. Now we proceed, putting boxes into boxes. This means that we create a second (1) order partition set to each first order partition in ℘3 . The union of all the (2) second order partition sets is denoted by ℘3 , and appears in the third column of Figure 1. 1 (0) (1) (2) S = ℘3 ℘3 ℘3 { abc {a,b,c} { abc , , c ab c ab c ab , , , ac ac b ac b b , , , a bc a bc a bc , , , a c a b c a b c b } , , a c b b c a a b c , , } (1) (2) Figure 1: The basic set S together with the partition sets ℘3 and ℘3 (1) (m) Definition. -
Fine Books in All Fields
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Notices of the American Mathematical
ISSN 0002-9920 Notices of the American Mathematical Society AMERICAN MATHEMATICAL SOCIETY Graduate Studies in Mathematics Series The volumes in the GSM series are specifically designed as graduate studies texts, but are also suitable for recommended and/or supplemental course reading. With appeal to both students and professors, these texts make ideal independent study resources. The breadth and depth of the series’ coverage make it an ideal acquisition for all academic libraries that of the American Mathematical Society support mathematics programs. al January 2010 Volume 57, Number 1 Training Manual Optimal Control of Partial on Transport Differential Equations and Fluids Theory, Methods and Applications John C. Neu FROM THE GSM SERIES... Fredi Tro˝ltzsch NEW Graduate Studies Graduate Studies in Mathematics in Mathematics Volume 109 Manifolds and Differential Geometry Volume 112 ocietty American Mathematical Society Jeffrey M. Lee, Texas Tech University, Lubbock, American Mathematical Society TX Volume 107; 2009; 671 pages; Hardcover; ISBN: 978-0-8218- 4815-9; List US$89; AMS members US$71; Order code GSM/107 Differential Algebraic Topology From Stratifolds to Exotic Spheres Mapping Degree Theory Matthias Kreck, Hausdorff Research Institute for Enrique Outerelo and Jesús M. Ruiz, Mathematics, Bonn, Germany Universidad Complutense de Madrid, Spain Volume 110; 2010; approximately 215 pages; Hardcover; A co-publication of the AMS and Real Sociedad Matemática ISBN: 978-0-8218-4898-2; List US$55; AMS members US$44; Española (RSME). Order code GSM/110 Volume 108; 2009; 244 pages; Hardcover; ISBN: 978-0-8218- 4915-6; List US$62; AMS members US$50; Ricci Flow and the Sphere Theorem The Art of Order code GSM/108 Simon Brendle, Stanford University, CA Mathematics Volume 111; 2010; 176 pages; Hardcover; ISBN: 978-0-8218- page 8 Training Manual on Transport 4938-5; List US$47; AMS members US$38; and Fluids Order code GSM/111 John C. -
Multiple Stirling Number Identities
Multiple Stirling Number Identities H. Coskun Department of Mathematics, Texas A&M University{Commerce, Binnion Hall, Room 314, Commerce, TX 75429 Abstract A remarkable multiple analogue of the Stirling numbers of the first and sec- ond kind was recently constructed by the author. Certain summation iden- tities, and related properties of this family of multiple special numbers are investigated in the present paper. Keywords: multiple binomial coefficients, multiple special numbers, qt-Stirling numbers, well-poised symmetric rational Macdonald functions 2000 MSC: 05A10, 11B65, 33D67 1. Introduction The Stirling numbers of the first and second kind are studied and their properties are investigated extensively in number theory and combinatorics. One dimensional generalizations of these numbers have also been subject of interest. An important class of generalizations is their one parameter q- extensions. Many have made significant contributions to such q-extensions investigating their properties and applications. We will give references to some of these important work in Section 3 below. In a recent paper, the author took a major step and constructed an el- egant multiple qt-generalization of Stirling numbers of the first and second kind, besides sequences of other special numbers including multiple bino- mial, Fibonacci, Bernoulli, Catalan, and Bell numbers [16]. In this paper, we focus on multiple Stirling numbers of both kinds, and give interesting new identities satisfied by them. Email address: [email protected] (H. Coskun) URL: http://faculty.tamuc.edu/hcoskun (H. Coskun) Preprint submitted to American Journal of Mathematics The multiple generalizations developed in [16] are given in terms of the qt-binomial coefficients constructed in the same paper.