DOCSLIB.ORG

Touchard Polynomials, Stirling Numbers and Random Permutations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Touchard Polynomials, Stirling Numbers and Random Permutations Touchard Polynomials, Stirling Numbers and Random Permutations Ross Pinsky Department of Mathematics, Technion 32000 Haifa, ISRAEL September 3, 2018 Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations jSnj = n! 1 2 3 4 5 6 S 3 σ = = (124)(36)(5), 6 1 2 4 6 1 5 3 1 2 3 4 5 6 S 3 σ = = (136245) = (624513) 6 2 3 4 6 5 1 2 σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6. There are (n − 1)! different n-cycles in Sn. Notation: [n] = f1; 2; ··· ; ng Sn = permutations of [n] Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations 1 2 3 4 5 6 S 3 σ = = (124)(36)(5), 6 1 2 4 6 1 5 3 1 2 3 4 5 6 S 3 σ = = (136245) = (624513) 6 2 3 4 6 5 1 2 σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6. There are (n − 1)! different n-cycles in Sn. Notation: [n] = f1; 2; ··· ; ng Sn = permutations of [n] jSnj = n! Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations 1 2 3 4 5 6 S 3 σ = = (136245) = (624513) 6 2 3 4 6 5 1 2 σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6. There are (n − 1)! different n-cycles in Sn. Notation: [n] = f1; 2; ··· ; ng Sn = permutations of [n] jSnj = n! 1 2 3 4 5 6 S 3 σ = = (124)(36)(5), 6 1 2 4 6 1 5 3 Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6. There are (n − 1)! different n-cycles in Sn. Notation: [n] = f1; 2; ··· ; ng Sn = permutations of [n] jSnj = n! 1 2 3 4 5 6 S 3 σ = = (124)(36)(5), 6 1 2 4 6 1 5 3 1 2 3 4 5 6 S 3 σ = = (136245) = (624513) 6 2 3 4 6 5 1 2 Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations There are (n − 1)! different n-cycles in Sn. Notation: [n] = f1; 2; ··· ; ng Sn = permutations of [n] jSnj = n! 1 2 3 4 5 6 S 3 σ = = (124)(36)(5), 6 1 2 4 6 1 5 3 1 2 3 4 5 6 S 3 σ = = (136245) = (624513) 6 2 3 4 6 5 1 2 σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6. Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Notation: [n] = f1; 2; ··· ; ng Sn = permutations of [n] jSnj = n! 1 2 3 4 5 6 S 3 σ = = (124)(36)(5), 6 1 2 4 6 1 5 3 1 2 3 4 5 6 S 3 σ = = (136245) = (624513) 6 2 3 4 6 5 1 2 σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6. There are (n − 1)! different n-cycles in Sn. Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations n X k ≡ ank x ; x 2 R; n ≥ 1: (1) k=1 Unsigned Stirling Numbers of the First Kind: js(n; k)j := the number of permutations in Sn with exactly k cycles; 1 ≤ k ≤ n Recurrence relation js(n; n)j = 1; js(n; 1)j = (n − 1)!; n ≥ 1; (2) js(n + 1; k)j = njs(n; k)j + js(n; k − 1)j; 2 ≤ k ≤ n: It is easy to check that the coefficients fank g in (1) also satisfy (2); thus, ank = js(n; k)j. Therefore n X x(n) = js(n; k)jxk : (3) k=1 Rising Factorials x(n) := x(x +1) ··· (x +n−1) Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Unsigned Stirling Numbers of the First Kind: js(n; k)j := the number of permutations in Sn with exactly k cycles; 1 ≤ k ≤ n Recurrence relation js(n; n)j = 1; js(n; 1)j = (n − 1)!; n ≥ 1; (2) js(n + 1; k)j = njs(n; k)j + js(n; k − 1)j; 2 ≤ k ≤ n: It is easy to check that the coefficients fank g in (1) also satisfy (2); thus, ank = js(n; k)j. Therefore n X x(n) = js(n; k)jxk : (3) k=1 Rising Factorials n (n) X k x := x(x +1) ··· (x +n−1) ≡ ank x ; x 2 R; n ≥ 1: (1) k=1 Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Recurrence relation js(n; n)j = 1; js(n; 1)j = (n − 1)!; n ≥ 1; (2) js(n + 1; k)j = njs(n; k)j + js(n; k − 1)j; 2 ≤ k ≤ n: It is easy to check that the coefficients fank g in (1) also satisfy (2); thus, ank = js(n; k)j. Therefore n X x(n) = js(n; k)jxk : (3) k=1 Rising Factorials n (n) X k x := x(x +1) ··· (x +n−1) ≡ ank x ; x 2 R; n ≥ 1: (1) k=1 Unsigned Stirling Numbers of the First Kind: js(n; k)j := the number of permutations in Sn with exactly k cycles; 1 ≤ k ≤ n Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations It is easy to check that the coefficients fank g in (1) also satisfy (2); thus, ank = js(n; k)j. Therefore n X x(n) = js(n; k)jxk : (3) k=1 Rising Factorials n (n) X k x := x(x +1) ··· (x +n−1) ≡ ank x ; x 2 R; n ≥ 1: (1) k=1 Unsigned Stirling Numbers of the First Kind: js(n; k)j := the number of permutations in Sn with exactly k cycles; 1 ≤ k ≤ n Recurrence relation js(n; n)j = 1; js(n; 1)j = (n − 1)!; n ≥ 1; (2) js(n + 1; k)j = njs(n; k)j + js(n; k − 1)j; 2 ≤ k ≤ n: Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Therefore n X x(n) = js(n; k)jxk : (3) k=1 Rising Factorials n (n) X k x := x(x +1) ··· (x +n−1) ≡ ank x ; x 2 R; n ≥ 1: (1) k=1 Unsigned Stirling Numbers of the First Kind: js(n; k)j := the number of permutations in Sn with exactly k cycles; 1 ≤ k ≤ n Recurrence relation js(n; n)j = 1; js(n; 1)j = (n − 1)!; n ≥ 1; (2) js(n + 1; k)j = njs(n; k)j + js(n; k − 1)j; 2 ≤ k ≤ n: It is easy to check that the coefficients fank g in (1) also satisfy (2); thus, ank = js(n; k)j. Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Rising Factorials n (n) X k x := x(x +1) ··· (x +n−1) ≡ ank x ; x 2 R; n ≥ 1: (1) k=1 Unsigned Stirling Numbers of the First Kind: js(n; k)j := the number of permutations in Sn with exactly k cycles; 1 ≤ k ≤ n Recurrence relation js(n; n)j = 1; js(n; 1)j = (n − 1)!; n ≥ 1; (2) js(n + 1; k)j = njs(n; k)j + js(n; k − 1)j; 2 ≤ k ≤ n: It is easy to check that the coefficients fank g in (1) also satisfy (2); thus, ank = js(n; k)j. Therefore n X x(n) = js(n; k)jxk : (3) k=1 Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Substituting −x for x in (4), we have n (−x)n = −x(−x −1) ··· (−x −n+1) = (−1) x(x +1) ··· (x +n−1) = (−1)nx(n). Substituting above −x for x, we have n n (n) from (3) n X k (x)n = (−1) (−x) = (−1) js(n; k)j(−x) = k=1 n (5) X (−1)n−k js(n; k)jxk : k=1 Define s(n; k) := (−1)n−k js(n; k)j. The s(n; k) are called the Stirling Numbers of the First Kind. From (5) we have n X k (x)n = s(n; k)x : (6) k=1 Falling Factorials (x)n := x(x − 1) ··· (x − n + 1); x 2 R; n ≥ 1: (4) Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations (−1)nx(x +1) ··· (x +n−1) = (−1)nx(n). Substituting above −x for x, we have n n (n) from (3) n X k (x)n = (−1) (−x) = (−1) js(n; k)j(−x) = k=1 n (5) X (−1)n−k js(n; k)jxk : k=1 Define s(n; k) := (−1)n−k js(n; k)j. The s(n; k) are called the Stirling Numbers of the First Kind. From (5) we have n X k (x)n = s(n; k)x : (6) k=1 Falling Factorials (x)n := x(x − 1) ··· (x − n + 1); x 2 R; n ≥ 1: (4) Substituting −x for x in (4), we have (−x)n = −x(−x −1) ··· (−x −n+1) = Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations = (−1)nx(n). Substituting above −x for x, we have n n (n) from (3) n X k (x)n = (−1) (−x) = (−1) js(n; k)j(−x) = k=1 n (5) X (−1)n−k js(n; k)jxk : k=1 Define s(n; k) := (−1)n−k js(n; k)j. The s(n; k) are called the Stirling Numbers of the First Kind. From (5) we have n X k (x)n = s(n; k)x : (6) k=1 Falling Factorials (x)n := x(x − 1) ··· (x − n + 1); x 2 R; n ≥ 1: (4) Substituting −x for x in (4), we have n (−x)n = −x(−x −1) ··· (−x −n+1) = (−1) x(x +1) ··· (x +n−1) Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations Substituting above −x for x, we have n n (n) from (3) n X k (x)n = (−1) (−x) = (−1) js(n; k)j(−x) = k=1 n (5) X (−1)n−k js(n; k)jxk : k=1 Define s(n; k) := (−1)n−k js(n; k)j.
Load more

Recommended publications
  • Multivariate Expansions Associated with Sheffer-Type Polynomials and Operators
    Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 1 (2006), No. 4, pp. 451-473 MULTIVARIATE EXPANSIONS ASSOCIATED WITH SHEFFER-TYPE POLYNOMIALS AND OPERATORS BY TIAN XIAO HE, LEETSCH C. HSU, AND PETER J.-S. SHIUE Abstract With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general ex- pansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solu- [ tion to the problem of the expansion of A(tˆ)f(g(t)) (a composi- tion of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory. 1. Introduction The purpose of this paper is to study the following expansion problem. Problem 1. Let tˆ = (t1,t2,...,tr), A(tˆ), g(t) = (g1(t1), g2(t2),..., gr(tr)) and f(tˆ) be any given formal power series over the complex number Cr ˆ ′ d field with A(0) = 1, gi(0) = 0 and gi(0) 6= 0 (i = 1, 2, . , r). We wish to find the power series expansion in tˆ of the composite function A(tˆ)f(g(t)). Received October 18, 2005 and in revised form July 5, 2006. d Communicated by Xuding Zhu. AMS Subject Classification: 05A15, 11B73, 11B83, 13F25, 41A58. Key words and phrases: Multivariate formal power series, multivariate Sheffer-type polynomials, multivariate Sheffer-type differential operators, multivariate weighted Stirling numbers, multivariate Riordan array pair, multivariate exponential polynomials.
    [Show full text]
  • A Note on Some Identities of New Type Degenerate Bell Polynomials
    mathematics Article A Note on Some Identities of New Type Degenerate Bell Polynomials Taekyun Kim 1,2,*, Dae San Kim 3,*, Hyunseok Lee 2 and Jongkyum Kwon 4,* 1 School of Science, Xi’an Technological University, Xi’an 710021, China 2 Department of Mathematics, Kwangwoon University, Seoul 01897, Korea; [email protected] 3 Department of Mathematics, Sogang University, Seoul 04107, Korea 4 Department of Mathematics Education and ERI, Gyeongsang National University, Gyeongsangnamdo 52828, Korea * Correspondence: [email protected] (T.K.); [email protected] (D.S.K.); [email protected] (J.K.) Received: 24 October 2019; Accepted: 7 November 2019; Published: 11 November 2019 Abstract: Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind. Keywords: Bell polynomials; partially degenerate Bell polynomials; new type degenerate Bell polynomials MSC: 05A19; 11B73; 11B83 1. Introduction Studies on degenerate versions of some special polynomials can be traced back at least as early as the paper by Carlitz [1] on degenerate Bernoulli and degenerate Euler polynomials and numbers. In recent years, many mathematicians have drawn their attention in investigating various degenerate versions of quite a few special polynomials and numbers and discovered some interesting results on them [2–9].
    [Show full text]
  • Recurrence Relation Lower and Upper Bounds Maximum Parity Simple
    From Wikipedia, the free encyclopedia In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or .[1] Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. Stirling numbers of the second kind are one of two kinds of Stirling numbers, the other kind being called Stirling numbers of the first kind (or Stirling cycle numbers). Mutually inverse (finite or infinite) triangular matrices can be formed from the Stirling numbers of each kind according to the parameters n, k. 1 Definition The 15 partitions of a 4-element set 2 Notation ordered in a Hasse diagram 3 Bell numbers 4 Table of values There are S(4,1),...,S(4,4) = 1,7,6,1 partitions 5 Properties containing 1,2,3,4 sets. 5.1 Recurrence relation 5.2 Lower and upper bounds 5.3 Maximum 5.4 Parity 5.5 Simple identities 5.6 Explicit formula 5.7 Generating functions 5.8 Asymptotic approximation 6 Applications 6.1 Moments of the Poisson distribution 6.2 Moments of fixed points of random permutations 6.3 Rhyming schemes 7 Variants 7.1 Associated Stirling numbers of the second kind 7.2 Reduced Stirling numbers of the second kind 8 See also 9 References The Stirling numbers of the second kind, written or or with other notations, count the number of ways to partition a set of labelled objects into nonempty unlabelled subsets.
    [Show full text]
  • Arxiv:2103.13478V2 [Math.NT] 11 May 2021 Solution of the Differential Equation Y = E , Special Values of Bell Polynomials
    Solution of the Differential Equation y(k) = eay, Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients Ronald Orozco L´opez Departamento de Matem´aticas Universidad de los Andes Bogot´a, 111711 Colombia [email protected] Abstract In this paper special values of Bell polynomials are given by using the power series solution of the equation y(k) = eay. In addition, complete and partial exponential autonomous functions, exponential autonomous polyno- mials, autonomous polynomials and (k, a)-autonomous coefficients are de- fined. Finally, we show the relationship between various numbers counting combinatorial objects and binomial coefficients, Stirling numbers of second kind and autonomous coefficients. 1 Introduction It is a known fact that Bell polynomials are closely related to derivatives of composi- tion of functions. For example, Faa di Bruno [5], Foissy [6], and Riordan [10] showed that Bell polynomials are a very useful tool in mathematics to represent the n-th derivative of the composition of functions. Also, Bernardini and Ricci [2], Yildiz et arXiv:2103.13478v2 [math.NT] 11 May 2021 al. [12], Caley [3], and Wang [13] showed the relationship between Bell polynomials and differential equations. On the other hand, Orozco [9] studied the convergence of the analytic solution of the autonomous differential equation y(k) = f(y) by using Faa di Bruno’s formula. We can then look at differential equations as a source for investigating special values of Bell polynomials. In this paper we will focus on finding special values of Bell polynomials when the vector field f(x) of the autonomous differential equation y(k) = f(y) is the exponential function.
    [Show full text]
  • Generalized Bell Polynomials and the Combinatorics of Poisson Central Moments
    Generalized Bell polynomials and the combinatorics of Poisson central moments Nicolas Privault∗ Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University SPMS-MAS-05-43, 21 Nanyang Link Singapore 637371 February 25, 2011 Abstract We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson dis- tribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2. Key words: Bell polynomials, Poisson distribution, central moments, Stirling num- bers. Mathematics Subject Classification: 11B73, 60E07. 1 Introduction The moments of the Poisson distribution are well-known to be connected to the com- binatorics of the Stirling and Bell numbers. In particular the Bell polynomials Bn(λ) satisfy the relation n Bn(λ) = Eλ[Z ]; n 2 IN; (1.1) where Z is a Poisson random variable with parameter λ > 0, and n X Bn(1) = S(n; c) (1.2) c=0 ∗[email protected] 1 is the Bell number of order n, i.e. the number of partitions of a set of n elements. In this paper we study the central moments of the Poisson distribution, and we show that they can be expressed using the number of partitions of a set into subsets of size at least 2, in connection with an extension of the Bell polynomials. Consider the above mentioned Bell (or Touchard) polynomials Bn(λ) defined by the exponential generating function 1 n t X t eλ(e −1) = B (λ); (1.3) n! n n=0 λ, t 2 IR, cf.
    [Show full text]
  • Contents What These Numbers Count Triangle Scheme for Calculations
    From Wikipedia, the free encyclopedia In combinatorial mathematics, the Bell numbers count the number of partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. Starting with B0 = B1 = 1, the first few Bell numbers are: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, ... (sequence A000110 in the OEIS). The nth of these numbers, Bn, counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. Outside of mathematics, the same number also counts the number of different rhyme schemes for n-line poems.[1] As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, Bn is the nth moment of a Poisson distribution with mean 1. Contents 1 What these numbers count 1.1 Set partitions 1.2 Factorizations 1.3 Rhyme schemes 1.4 Permutations 2 Triangle scheme for calculations 3 Properties 3.1 Summation formulas 3.2 Generating function 3.3 Moments of probability distributions 3.4 Modular arithmetic 3.5 Integral representation 3.6 Log-concavity 3.7 Growth rate 4 Bell primes 5History 6 See also 7Notes 8 References 9 External links What these numbers count Set partitions In general, Bn is the number of partitions of a set of size n.
    [Show full text]
  • Compound Polynomials 65
    Polynomial maps with applications to combinatorics and probability theory by Dan Port B.S. Mathematics, University of California (1989) Submitted to the Department of Applied Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1994 ( Dan Port, 1993. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part, and to grant others the right to do so. -0c? 14 Author. ....... g "- . ~~~ ............. ............... /,!·'-Dbartment o d ematics ber 15, 1993 Certifiedby....... f// Certified by .... Chairman, Applied Mathematics Accepted by. .4.-.-. MAssACHusErrS nIVSmru-E David Vogan n'-r,4+,L0GY Chairman, Departmental Graduate Committee APR 25 1994 LIBRARES ^C.1enco, Polynomial maps with applications to combinatorics and probability theory by Dan Port Submitted to the Department of Applied Mathematics on October 15, 1993, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract. This work considers broad classes of polynomial maps which generalize the exponential Bell polynomials. These maps correspond to various convolutions (such as Hadamard and Cauchy convolutions) and have been extensively studied in combinatorics, but relatively little in connection with probability theory. It is shown that the exponential Bell Polynomials Y (xl, X2 , .. ) map the space of moments M to itself, a property which is called MP. The inverse image of M under (Yn) is the space KCof cumulants. By Hamburger's solution of the problem of moments, it follows that if the the Hankel determinants of the x, are non-negative, then so are those of the Yn.
    [Show full text]
  • The Generalized Touchard Polynomials Revisited Applied
    Applied Mathematics and Computation 219 (2013) 9978–9991 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc The generalized Touchard polynomials revisited ⇑ Toufik Mansour a, , Matthias Schork b a Department of Mathematics, University of Haifa, 31905 Haifa, Israel b Camillo-Sitte-Weg 25, 60488 Frankfurt, Germany article info abstract Keywords: We discuss the generalized Touchard polynomials introduced recently by Dattoli et al. as Touchard polynomial well as their extension to negative order introduced by the authors with operational meth- Stirling number ods. The connection to generalized Stirling and Bell numbers is elucidated and analogs to Bell number Burchnall’s identity are derived. A recursion relation for the generalized Touchard polyno- Generating function mials is established and it is shown that one can interpret some of the resulting formulas as binomial theorems for particular noncommuting variables. We suggest to generalize the generalized Touchard polynomials still further and introduce so called Comtet–Touchard functions which are associated to the powers of an arbitrary derivation. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction The Touchard polynomials (also called exponential polynomials) may be defined (see, e.g., [1,2,6,16,17]) for n 2 N by d n T ðxÞ :¼ eÀx x ex: ð1:1Þ n dx d In the following we also use the notation D ¼ @x ¼ dx. If we further denote the operator of multiplication with x by X, i.e., ðXf ÞðxÞ¼xf ðxÞ for allP functions f considered, we can make contact with operational formulas [16,17]. For example, using n n k k the fact that ðXDÞ ¼ k¼0Sðn; kÞX D , where Sðn; kÞ denotes the Stirling numbers of the second kind, one obtains directly from the definition of the Touchard polynomials the relation Xn k TnðxÞ¼ Sðn; kÞx ¼ BnðxÞ; ð1:2Þ k¼0 where the second equality corresponds to the definition of the conventional Bell polynomials.
    [Show full text]
  • Some Identities for Umbral Calculus Associated with Partially Degenerate Bell Numbers and Polynomials
    Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Some identities for umbral calculus associated with partially degenerate Bell numbers and polynomials Taekyun Kima,b, Dae San Kimc, Hyuck-In Kwonb, Seog-Hoon Rimd,∗ aDepartment of Mathematics, College of Science Tianjin Polytechnic University, Tianjin 300160, China. bDepartment of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. cDepartment of Mathematics, Sogang University, Seoul 121-742, Republic of Korea. dDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea. Communicated by Y. J. Cho Abstract In this paper, we study partially degenerate Bell numbers and polynomials by using umbral calculus. We give some new identities for these numbers and polynomials which are associated with special numbers and polynomial. c 2017 All rights reserved. Keywords: Partially degenerate Bell polynomials, umbral calculus. 2010 MSC: 05A19, 05A40, 11B83. 1. Introduction The Bell polynomials (also called the exponential polynomials and denoted by φn(x)) are defined by the generating function (see [4, 10, 16]) n x(et-1) t e = Beln(x) . (1.1) 1 n! n= X0 From (1.1), we note that 2 3 2 Bel0(x) = 1, Bel1(x) = x, Bel2(x) = x + x, Bel3(x) = x + 3x + x, 4 3 2 5 4 3 2 Bel4(x) = x + 6x + 7x + x, Bel5(x) = x + 10x + 25x + 15x + x, ··· . When x = 1, Beln = Beln(1) are called the Bell numbers. As is well-known, the Stirling numbers of the second kind are defined by the generating function (see [12, 14, 16]) n t m t e - 1 = m! S2(n, m) , (m > 0).
    [Show full text]
  • Impact of Maternal Effects on Ranking of Animal Models for Genetic
    Scientific Journal of Pure and Applied Sciences (2013) 2(12) 401-460 ISSN 2322-2956 doi: 10.14196/sjpas.v2i12.1117 Contents lists available at Sjournals Journal homepage: www.Sjournals.com Original article Definiability of combinatorial functions and their linear recurrence relationships within a polylogarithmic triangularizable matrix employing surjective bilipschitz functions and other isomorphisms of metric spaces for forecasting seasonal endemic onchocerciasis transmission zones in Burkina Faso B.G. Jacoba,*, R.J. Novaka, L. Toeb, M.S. Sanfob, S. Caliskhana, R. Tingueriae, A. Pared, M. Nomab, L. Yameogoc, T.R. Unnascha aGlobal Infectious Disease Research Program, Department of Public Health, College of Public Health, University of South Florida, 3720 Spectrum Blvd, Suite 304, Tampa, Florida, USA 33612. bMultiDisease Surveillance Centre (MDSC) 1473, Avenue Naba Zombre, 01 B.P 549 Ouagadougou 01 Burkina Faso. cAfrican Programme for Onchocerciasis Control (APOC)1473, Avenue Naba Zombre, 01 B.P 549 Ouagadougou 01 Burkina Faso. dMinistere de la Sante Ouagadougou Burkina Faso. eSchool of Economic, Political and Policy Sciences. The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX 75080-302. *Corresponding author; Global Infectious Disease Research Program, Department of Public Health, College of Public Health, University of South Florida, 3720 Spectrum Blvd, Suite 304, Tampa, Florida, USA 33612. A R T I C L E I N F O A B S T R A C T Article history: In this research, prevalence values based on Monthly Biting Received 02 November 2013 Rates (MBR) were employed as a response variable in a Poisson Accepted 20 November 2013 probability model framework for quantitatively regressing multiple Available online 31 December 2013 georefernced explanatory environmental-related explanatory covariates of seasonally-sampled larval habitat of Similium Keywords: damnosum s.l.a black fly vector of Onchocerciasis in a riverine study Prevalence site in Burkina Faso.
    [Show full text]
  • Arxiv:2106.12935V1 [Math.CO]
    (p,q)-ANALOGUES OF THE GENERALIZED TOUCHARD POLYNOMIALS AND STIRLING NUMBERS LAHCEN OUSSI Abstract. In this paper we introduce a (p, q)-deformed analogues of the generalized Touchard polynomials and Stirling numbers, the post-quantum analogues of the q-deformed generalized Touchard polynomials and Stirling numbers. The connection between these deformations is established. A recurrence relation for the (p, q)-deformed generalized Touchard polynomials is expounded, elucidating a (p, q)-deformation of Spivey’s relation. 1. Introduction The Stirling numbers of the second kind, denoted by S(n, k), count the number of ways to partition a set of size n into k non-empty subsets. They appear as coefficients in the normal d n ordering of X dx as follows n d n d k (1) X = S(n, k)Xk , dx dx k X=0 where X denotes the operator of multiplication with the variable, that is, Xf(x) = xf(x), d and dx is the derivative operator with respect to x. The relation (1) has deep connection in the physical literature, which was rediscovered by Katriel [20]. Namely, it is connected with the creation operator a and the annihilation operator a† satisfying the commutation relation aa† a†a = 1 in the Boson Fock space [1, 27]. Moreover,− the Stirling numbers of the second kind S(n, k), appear also as coefficients in the expansion of n k−1 xn = S(n, k) (x j), − k j X=0 Y=0 and they satisfy the recurrence formula (2) S(n, k)= S(n 1, k 1) + S(n 1, k).
    [Show full text]
  • SOME INEQUALITIES for GENERALIZED BELL-TOUCHARD POLYNOMIALS 647 Where Sr(N+R,I+R) Are the R-Stirling Numbers of the Second Kind [2]
    Journal of Mathematical Inequalities Volume 13, Number 3 (2019), 645–654 doi:10.7153/jmi-2019-13-42 SOME INEQUALITIES FOR GENERALIZED BELL–TOUCHARD POLYNOMIALS ∗ HAI-RONG YAN,QIAO-LING ZHANG AND AI-MIN XU (Communicated by N. Elezovi´c) Abstract. A unified generalization for the Bell-Touchard polynomials of order k and the r -Bell polynomials is established. It is shown that the generating function of the generalized Bell- Touchard polynomials is logarithmically absolutely monotonic. Applying this result we obtain some inequalities for the generalized Bell-Touchard polynomials. In particular, we obtain the logarithmic convexity of the generalized Bell-Touchard polynomials. 1. Introduction and main results Asai et al. [1] introduced the Bell number of order k as follows. For an integer ( ) k 1, define the k-times iterated exponential function denoted by expk z : ( )= ( ···( ( ))). expk z exp exp exp z (1.1) k−times { ( )}∞ ( ) Let Bk n n=0 be the sequence of numbers given in the power series of expk z , namely, ∞ ( ) ( )= Bk n n. expk z ∑ z (1.2) n=0 n! { ( )}∞ The Bell numbers bk n n=0 of order k are defined by ( ) ( )= Bk n , . bk n ( ) n 0 (1.3) expk 0 In particular, when k = 2, the numbers b2(n) are usually known as the Bell numbers, the first few terms of which are 1,1,2,5,15,52,203. Thus, it is natural that ∞ z− b (n) ee 1 = ∑ 2 zn (1.4) n=0 n! Mathematics subject classification (2010): 11B73, 26A48, 26A51, 33B10. Keywords and phrases: Bell-Touchard polynomial, inequality, absolutely monotonic, completely monotonic, logarithmic convexity.
    [Show full text]