DOCSLIB.ORG

Sequences of Binomial Type with Persistent Roots

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sequences of Binomial Type with Persistent Roots JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 199, 39]58Ž. 1996 ARTICLE NO. 0125 Sequences of Binomial Type with Persistent Roots A. Di Bucchianico* Department of Mathematics and Computing Science, Eindho¨en Uni¨ersity of Technology, P.O. Box 513, 5600 MB Eindho¨en, The Netherlands and View metadata, citation and similar papers at core.ac.uk brought to you by CORE D. E. Loeb² provided by Elsevier - Publisher Connector LaBRI, Uni¨ersite de Bordeaux I, 33405 Talence, France Submitted by George Gasper Received March 22, 1995 Ž. Ž Ž. We find all sequences of polynomials pnnG0with persistent roots i.e., pxn s Ž.Ž.Ž.. cxn yrx12yr??? x y rnthat are of binomial type in Viskov's generalization of Rota's umbral calculus to generalized Appell polynomials. We show that such sequences only exist in the classical umbral calculus, the divided difference umbral 2 calculus, and the new ``hyperbolic'' umbral calculus generated by Ž.drdx' .In each of these three umbral calculi, we also find all Sheffer sequences with persistent roots. Q 1996 Academic Press, Inc. 1. INTRODUCTION Most sequences of polynomials that interest mathematicians fall into one of the following three categories or their generalizations: v sequences of binomial typewx 13]15 , v sequences with persistent rootswx 4, 5 and v sequences of orthogonal polynomialswx 3 . These theories are very rich; powerful formulas allow the computation of one such sequence of polynomials in terms of another of the same type. *Author supported by NATO CRG 930554. WWW URL http:rrwww.win.tue.nlrwinr mathrbsrstatisticsrbucchianico. E-mail address: [email protected]. ²Author partially supported by URA CNRS 1304, EC Grant CHRX-CT93-0400, the PRC Maths-Info, and NATO CRG 930554. WWW URL http:rrwww.labri.u-bordeaux.frr;loeb. E-mail address: [email protected]. 39 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 40 DI BUCCHIANICO AND LOEB However, to go from, for example, an arbitraryŽ. generalized convolution sequence to an arbitrary sequence with persistent roots, it is necessary to n pass through a sequence that is both, for example, x rn!. This gives rise to the following three questions: What are the con¨olution sequences of orthogonal polynomials? This first question is treated bywx 1, 3, 10, 16 . What are the persistent sequences of orthogonal polynomials? Since persistent sequences obey a two-term recurrence and orthogonal se- quences obey a three-term recurrenceŽ. Favard's theorem , there are no persistent sequences of orthogonal polynomials. What are the persistent sequences of binomial type? The objective of this paper is the complete resolution of this last question. Up to rescaling, all persistent convolution sequences can be found in Table I. Ž. A sequence of polynomials pnnG0is said to be of binomial type if it obeys the identity n n Epy Ž. x pxpyŽ. Ž.,1Ž. nnsÝž/k ykk ks0 y where EpxŽ.spx Žqy .. Well-known examples include the powers of x ŽŽ..in which case Eq. 1 is called the binomial theorem , the Abel polynomi- n1 als xxŽ.yna y, the Laguerre polynomials, and so onwx 13, 14 . TABLE I Complete List of Persistent Convolution Sequences Umbral calculus Roots Delta operator Polynomials Name bnnF rfŽD b .Ž.px qnŽ.x xn any any any 0, 0, 0, 0, . Dpxb Ž. bn 1 xpŽ. x p Ž.1 divided y n 1 1 0,1,1,1,... xxŽ.y1 y difference 1 y txy1 Ž.Ž. x classical n! exp 0, 1, 2, 3, . pxq1ypx ž/n 2 2 xxŽ.y1??? Ž.x y Ž.n y 1 Ž. 1 hyperbolic 2n ! cosh 0, 1, 4, 9, . 2px'q1q Ž.wx Ž.2n! 2 pxŽ.wx'y1y2pxŽ. SEQUENCES WITH PERSISTENT ROOTS 41 Ž. Ž. If pnnG0 is of binomial type, then the sister sequence qnnG0s Ž. pnnrn!G0 obeys the identity n y EqnnŽ. x sÝqxqyykk Ž. Ž.. ks0 It is often more convenient to consider the sister con¨olution sequence rather than the sequence of binomial type itself. An important generalization of the umbral calculus involves replacing n the derivative D with a homogeneous linear operator of the form Dxb s ny1 Ž. y bxnnrby1where bn/ 0 for all n provided that the shift E yFŽ.F Ž.Ý` n is replaced with Ebbs yD where t s ns0t rbnwx11, 12, 20, 21 . The resulting polynomials are sometimes called generalized Appell poly- nomials. However, umbral calculus does not include all important sequences of polynomials even as generalized above. For that reason, we now turn to sequences with persistent roots. Ž. A sequence pnnG0 is said to have persistent roots or be persistent if Ž. Ž. pxnndivides pxq1for all n. That is to say, all of the roots in pnpersist in pnq1. Ž. In Section 3, we consider the classical umbral calculus bns n!. All persistent convolution sequences have their roots in an arithmetic pro- gression pxnŽ.sxx Žya .??? Ž.x y Žn y 1 .a . Moreover, if we allow Sheffer sequences, then we are no longer limited to progressions which start at zero. We then consider two other umbral calculi and characterize all of their convolution and Sheffer sequences with persistent roots. In Section 4, we Ž. consider the divided difference umbral calculus bn s 18,17wx]19 which is important in interpolation theory. In Section 5, we consider a new ŽŽ.. umbral calculus bn s 2n ! which we call the hyperbolic umbral calculus since FŽ.t s cosh Ž't .. Surprisingly, these three examples give essentially all persistent convolu- Ž Ž .. Ž nn . tion sequences. Obviously, any sequence qxnnG0scxrbnnG0 has persistent rootsŽ. all zero , and is the basic convolution sequence for the delta operator Dbrc. However, in Section 6, we show that, other than these trivial examples, all persistent convolution sequences belong to the classical, divided difference, or hyperbolic umbral calculus. We have thus explicitly characterized all persistent divided difference polynomial se- quences. Our work was assisted by extensive use of the computer algebra system Maple and our umbral calculus packagewx 2 . 42 DI BUCCHIANICO AND LOEB 2. GENERATING FUNCTIONS We show in this section that in any umbral calculus, there is, up to vertical and horizontal scaling, at most one nontrivial persistent convolu- tion sequence. These results are superseded by the explicit examples in Sections 3]5, and the completeness results in Section 6 which follow as consequences of these uniqueness results. 2.1. Umbral Calculus Modern umbral calculus employs powerful operator methods to derive results on polynomialswx 13]15 . Different commutation classes of operators give yield to analogous results concerning different sequences of polynomialswx 9, 21 . The classical umbral calculus concerns the family of operators that commute with the derivative D. These operators can all be expressed as formal power series in the derivative. Thus, they all commute with each y other. In particular, they commute with the shift operator E s expŽ.yD . Hence, these operators are usually called shift-invariant operators. The situation in other umbral calculi is analogous. A generalized deri¨a- ti¨e is defined by a non-zero generalized factorial function bn as bn ¡ x ny1 if n ) 0, and Dxn b s~ bny1 ¢0ifns0. y Ž. Ž. The generalized shift operator is then Ebbs F yD where F t s Ý`n ns0 trbn so that we have the following generalized binomial theorem wx13, 20 n bn yn knyk Exb sÝ xy . bb ks0 knyk 2.2. Con¨olution Sequences Ž. The sequences of polynomials pnnG0under consideration in the um- Ž. bral calculus are required to satisfy deg pn s n. These sequences are studied in terms of the operators which define them. For example, let f be a delta series, i.e., a formal power series with zero Ž. Ž . Ž. Ž. constant term. Then any sequence qnnG0satisfying fDbn q xsqx ny1 Ž. will be said to be Sheffer. If further we have qn 0 s 0 for n ) 0, and Ž. q0 '1, then qnnG0is said to be a convolution sequence. SEQUENCES WITH PERSISTENT ROOTS 43 In the classical umbral calculus, convolution sequences qn have generat- ing functions of the form ` n Žy1. ÝqxtnŽ. sexpŽ.xfŽ. t , ns0 where f Žy1. denotes the compositional inverse of f. In general, convolution sequences qn are characterized by generating functions of the form ` n Žy1. ÝqxtnŽ. sFŽ.xfŽ. t .2Ž. ns0 The delta operator of qxnbŽ.is then fD Ž .. Ž. Thus, each convolution sequence qnnG0is determined uniquely by an umbral calculus b and a delta series f. Conversely, every convolution sequence belongs to essentially only one umbral calculus b Ž.Lemma 6.1 . Ž. However, if we require that qnnG0have persistent roots, then we can Ž. say much more. In this case, qnnG0is determined by the umbral calculus band the polynomial q2 alone. ROPOSITION Ž. P 2.1. Let qnnG0 be a persistent con¨olution sequence in the umbral calculus b with respect to the delta operator fŽ. Dbn with roots r . Then either: nn v Ž. Ž. The sequence is tri¨ial, that is, qxnnscxrb,fDbbsDrc,and rn s0for all n G 1, or else r v Ž. Ž2 . Ž Ž. r12s0, r / 0, and f Dbbs cE y1rr22. For c s r s 1, fDb is called the forward difference operator D b.. We will give two proofs of this important result. The first one uses operator methods, and the second one uses generating functions. r 2 Ž. Proof 1. Case r2/ 0. By the generalized binomial theorem, Eqbn xs Ýn Ž.Ž. Ž. Ž. ks0qxqrnykk2. However, qrk21s0 for k G 2, and qxsxrc.
Load more

Recommended publications
  • Relationship of Bell's Polynomial Matrix and K-Fibonacci Matrix
    American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Relationship of Bell’s Polynomial Matrix and k-Fibonacci Matrix Mawaddaturrohmaha, Sri Gemawatib* a,bDepartment of Mathematics, University of Riau, Pekanbaru 28293, Indonesia aEmail: [email protected] bEmail: [email protected] Abstract The Bell‟s polynomial matrix is expressed as , where each of its entry represents the Bell‟s polynomial number.This Bell‟s polynomial number functions as an information code of the number of ways in which partitions of a set with certain elements are arranged into several non-empty section blocks. Furthermore, thek- Fibonacci matrix is expressed as , where each of its entry represents the k-Fibonacci number, whose first term is 0, the second term is 1 and the next term depends on a positive integer k. This article aims to find a matrix based on the multiplication of the Bell‟s polynomial matrix and the k-Fibonacci matrix. Then from the relationship between the two matrices the matrix is obtained. The matrix is not commutative from the product of the two matrices, so we get matrix Thus, the matrix , so that the Bell‟s polynomial matrix relationship can be expressed as Keywords: Bell‟s Polynomial Number; Bell‟s Polynomial Matrix; k-Fibonacci Number; k-Fibonacci Matrix. 1. Introduction Bell‟s polynomial numbers are studied by mathematicians since the 19th century and are named after their inventor Eric Temple Bell in 1938[5] Bell‟s polynomial numbers are represented by for each n and kelementsof the natural number, starting from 1 and 1 [2, p 135] Bell‟s polynomial numbers can be formed into Bell‟s polynomial matrix so that each entry of Bell‟s polynomial matrix is Bell‟s polynomial number and is represented by [11].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • A FAMILY of SEQUENCES of BINOMIAL TYPE 3 and If Jp +1= N Then This Difference Is 0
    A FAMILY OF SEQUENCES OF BINOMIAL TYPE WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ Abstract. For delta operator aD bDp+1 we find the corresponding polynomial se- − 1 2 quence of binomial type and relations with Fuss numbers. In the case D 2 D we show that the corresponding Bessel-Carlitz polynomials are moments of the− convolu- tion semigroup of inverse Gaussian distributions. We also find probability distributions νt, t> 0, for which yn(t) , the Bessel polynomials at t, is the moment sequence. { } 1. Introduction ∞ A sequence wn(t) n=0 of polynomials is said to be of binomial type (see [10]) if deg w (t)= n and{ for} every n 0 and s, t R we have n ≥ ∈ n n (1.1) w (s + t)= w (s)w − (t). n k k n k Xk=0 A linear operator Q of the form c c c (1.2) Q = 1 D + 2 D2 + 3 D3 + ..., 1! 2! 3! acting on the linear space R[x] of polynomials, is called a delta operator if c1 = 0. Here D denotes the derivative operator: D1 := 0 and Dtn := n tn−1 for n 1. We6 will write Q = g(D), where · ≥ c c c (1.3) g(x)= 1 x + 2 x2 + 3 x3 + .... 1! 2! 3! There is one-to-one correspondence between sequences of binomial type and delta ∞ operators, namely if Q is a delta operator then there is unique sequence w (t) of { n }n=0 binomial type satisfying Qw0(t) = 0 and Qwn(t)= n wn−1(t) for n 1.
    [Show full text]
  • Some New Identities Involving Sheffer-Appell Polynomial Sequences Via Matrix Approach
    SOME NEW IDENTITIES INVOLVING SHEFFER-APPELL POLYNOMIAL SEQUENCES VIA MATRIX APPROACH MOHD SHADAB, FRANCISCO MARCELLAN´ AND SAIMA JABEE∗ Abstract. In this contribution some new identities involving Sheffer-Appell polyno- mial sequences using generalized Pascal functional and Wronskian matrices are de- duced. As a direct application of them, identities involving families of polynomials as Euler, Bernoulli, Miller-Lee and Apostol-Euler polynomials, among others, are given. 1. introduction Sequences of polynomials play an important role in many problems of pure and applied mathematics in the framework of approximation theory, statistics, combinatorics and classical analysis (see, for example, [19, 22{25]). The sequence of Sheffer polynomials constitutes one of the most important family of polynomial sequences. A polynomial sequence fsn(x)gn≥0 is said to be a Sheffer polynomial sequence [6, 9, 24, 27] if its generating function has the following form: 1 X yn A(y)exH(y) = s (x) ; (1.1) n n! n=0 where A(y) = A0 + A1y + ··· ; and 2 H(y) = H1y + H2y + ··· ; with A0 6= 0 and H1 6= 0. Let us recall an alternative definition of the Sheffer polynomial sequences [24, Pg. 17]. P1 yn P1 yn Indeed, let h(y) = n=1 hn n! ; h1 6= 0; and l(y) = n=0 ln n! , l0 6= 0; be, respectively, delta series and invertible series with complex coefficients. Then there exists a unique sequence of Sheffer polynomials fsn(x)gn≥0 satisfying the orthogonality conditions k hl(y)h(y) jsn(x)i = n!δn;k 8 n; k = 0; (1.2) where δn;k is the Kronecker delta.
    [Show full text]
  • Fitting and Graphing Discrete Distributions
    Chapter 3 Fitting and graphing discrete distributions {ch:discrete} Discrete data often follow various theoretical probability models. Graphic displays are used to visualize goodness of fit, to diagnose an appropriate model, and determine the impact of individual observations on estimated parameters. Not everything that counts can be counted, and not everything that can be counted counts. Albert Einstein Discrete frequency distributions often involve counts of occurrences of events, such as ac- cident fatalities, incidents of terrorism or suicide, words in passages of text, or blood cells with some characteristic. Often interest is focused on how closely such data follow a particular prob- ability distribution, such as the binomial, Poisson, or geometric distribution, which provide the basis for generating mechanisms that might give rise to the data. Understanding and visualizing such distributions in the simplest case of an unstructured sample provides a building block for generalized linear models (Chapter 9) where they serve as one component. The also provide the basis for a variety of recent extensions of regression models for count data (Chapter ?), allow- ing excess counts of zeros (zero-inflated models), left- or right- truncation often encountered in statistical practice. This chapter describes the well-known discrete frequency distributions: the binomial, Pois- son, negative binomial, geometric, and logarithmic series distributions in the simplest case of an unstructured sample. The chapter begins with simple graphical displays (line graphs and bar charts) to view the distributions of empirical data and theoretical frequencies from a specified discrete distribution. It then describes methods for fitting data to a distribution of a given form and simple, effective graphical methods than can be used to visualize goodness of fit, to diagnose an appropriate model (e.g., does a given data set follow the Poisson or negative binomial?) and determine the impact of individual observations on estimated parameters.
    [Show full text]
  • Bell Polynomials in Combinatorial Hopf Algebras Ammar Aboud, Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet
    Bell polynomials in combinatorial Hopf algebras Ammar Aboud, Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet To cite this version: Ammar Aboud, Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet. Bell polynomials in combinatorial Hopf algebras. 2015. hal-00945543v2 HAL Id: hal-00945543 https://hal.archives-ouvertes.fr/hal-00945543v2 Preprint submitted on 8 Apr 2015 (v2), last revised 21 Jan 2016 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bell polynomials in combinatorial Hopf algebras Ammar Aboud,∗ Jean-Paul Bultel,y Ali Chouriaz Jean-Gabriel Luquexand Olivier Mallet{ April 3, 2015 Keywords: Bell polynomials, Symmetric functions, Hopf algebras, Faà di Bruno algebra, Lagrange inversion, Word symmetric functions, Set partitions. Abstract Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities etc. Many of the formulæ on Bell polynomials involve combinatorial objects (set partitions, set partitions into lists, permutations etc). So it seems natural to investigate analogous formulæ in some combinatorial Hopf algebras with bases indexed with these objects. In this paper we investigate the connexions between Bell polynomials and several combinatorial Hopf algebras: the Hopf algebra of symmetric functions, the Faà di Bruno algebra, the Hopf algebra of word symmetric functions etc.
    [Show full text]
  • A New Formula of Q-Fubini Numbers Via Goncharov Polynomials
    A NEW FORMULA OF q-FUBINI NUMBERS VIA GONC˘AROV POLYNOMIALS Adel Hamdi 20 aoˆut 2019 Faculty of Science of Gabes, Department of Mathematics, Cit´eErriadh 6072, Zrig, Gabes, Tunisia Abstract Connected the generalized Gon˘carov polynomials associated to a pair (∂, Z) of a delta operator ∂ and an interpolation grid Z, introduced by Lorentz, Tringali and Yan in [7], with the theory of binomial enumeration and order statistics, a new q-deformed of these polynomials given in this paper allows us to derive a new combinatorial formula of q-Fubini numbers. A combinatorial proof and some nice algebraic and analytic properties have been expanded to the q-deformed version. Keywords: q-delta operators, polynomials of q-binomial type, Gon˘carov polynomials, order partitions, q-Fubini numbers. 2010 Mathematics Subject Classification. 05A10, 41A05, 05A40. 1 Introduction This paper grew out of the recent work, generalized Gon˘carov polynomials, of Lo- rentz, Tringali and Yan in [7] where these polynomials are seen as a basis of solutions for the Interpolation problem : Find a polynomial f(x) of degree n such that the ith delta operator ∂ of f(x) at a given complex number ai has value bi, for i = 0, 1, 2, .... There is a natural q-analog of this interpolation by replacing the delta operator with a q-delta arXiv:1908.06939v1 [math.CO] 19 Aug 2019 operator, we extend these polynomials into a generalized q-Gon˘carov basis (tn,q(x))n≥0, i N defined by the q-biorthogonality relation εzi (∂q(tn,q(x))) = [n]q!δi,n, for all i, n ∈ , where Z = (zi)i≥0 is a sequence of scalars and εzi the evaluation at zi.
    [Show full text]
  • Pp. 101–130. the Classical Umbral Calculus
    Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 8(2009), pp. 101–130. The classical umbral calculus: She↵er sequences Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato Abstract1. Following the approach of Rota and Taylor [17], we present an innovative theory of She↵er sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simplifications and conceptual clarifica- tions in many results involving She↵er sequences. To give an indication of the e↵ectiveness of the theory, we describe applications to the well-known connection constants problem, to Lagrange inversion formula and to solving some recurrence relations. 1. Introduction As well known, many polynomial sequences like Laguerre polynomials, first and second kind Meixner polynomials, Poisson-Charlier polynomials and Stirling poly- nomials are She↵er sequences. She↵er sequences can be considered the core of umbral calculus: a set of tricks extensively used by mathematicians at the begin- ning of the twentieth century. Umbral calculus was formalized in the language of the linear operators by Gian- Carlo Rota in a series of papers (see [15], [16], and [14]) that have produced a plenty of applications (see [1]). In 1994 Rota and Taylor [17] came back to foundation of umbral calculus with the aim to restore, in light formal setting, the computational power of the original tools, heuristical applied by founders Blissard, Cayley and Sylvester. In this new setting, to which we refer as the classical umbral calculus, there are two basic devices. The first one is to represent a unital sequence of num- bers by a symbol ↵, called an umbra, that is, to represent the sequence 1,a1,a2,..
    [Show full text]
  • The Umbral Calculus of Symmetric Functions
    Advances in Mathematics AI1591 advances in mathematics 124, 207271 (1996) article no. 0083 The Umbral Calculus of Symmetric Functions Miguel A. Mendez IVIC, Departamento de Matematica, Caracas, Venezuela; and UCV, Departamento de Matematica, Facultad de Ciencias, Caracas, Venezuela Received January 1, 1996; accepted July 14, 1996 dedicated to gian-carlo rota on his birthday 26 Contents The umbral algebra. Classification of the algebra, coalgebra, and Hopf algebra maps. Families of binomial type. Substitution of admissible systems, composition of coalgebra maps and plethysm. Umbral maps. Shift-invariant operators. Sheffer families. Hammond derivatives and umbral shifts. Symmetric functions of Schur type. Generalized Schur functions. ShefferSchur functions. Transfer formulas and Lagrange inversion formulas. 1. INTRODUCTION In this article we develop an umbral calculus for the symmetric functions in an infinite number of variables. This umbral calculus is analogous to RomanRota umbral calculus for polynomials in one variable [45]. Though not explicit in the RomanRota treatment of umbral calculus, it is apparent that the underlying notion is the Hopf-algebra structure of the polynomials in one variable, given by the counit =, comultiplication E y, and antipode % (=| p(x))=p(0) E yp(x)=p(x+y) %p(x)=p(&x). 207 0001-8708Â96 18.00 Copyright 1996 by Academic Press All rights of reproduction in any form reserved. File: 607J 159101 . By:CV . Date:26:12:96 . Time:09:58 LOP8M. V8.0. Page 01:01 Codes: 3317 Signs: 1401 . Length: 50 pic 3 pts, 212 mm 208 MIGUEL A. MENDEZ The algebraic dual, endowed with the multiplication (L V M| p(x))=(Lx }My| p(x+y)) and with a suitable topology, becomes a topological algebra.
    [Show full text]
  • On the Combinatorics of Cumulants
    Journal of Combinatorial Theory, Series A 91, 283304 (2000) doi:10.1006Âjcta.1999.3017, available online at http:ÂÂwww.idealibrary.com on On the Combinatorics of Cumulants Gian-Carlo Rota Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Jianhong Shen School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 E-mail: jhshenÄmath.umn.edu View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by the Managing Editors provided by Elsevier - Publisher Connector Received July 26, 1999 dedicated to the memory of gian-carlo rota We study cumulants by Umbral Calculus. Various formulae expressing cumulants by umbral functions are established. Links to invariant theory, symmetric functions, and binomial sequences are made. 2000 Academic Press 1. INTRODUCTION Cumulants were first defined and studied by Danish scientist T. N. Thiele. He called them semi-invariants. The importance of cumulants comes from the observation that many properties of random variables can be better represented by cumulants than by moments. We refer to Brillinger [3] and Gnedenko and Kolmogorov [4] for further detailed probabilistic aspects on this topic. Given a random variable X with the moment generating function g(t), its nth cumulant Kn is defined as d n Kn= n log g(t). dt } t=0 283 0097-3165Â00 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. 284 ROTA AND SHEN That is, m K : n tn= g(t)=exp : n tn ,(1) n0 n! \ n1 n! + where, mn is the nth moment of X. Generally, if _ denotes the standard deviation, then 2 2 K1=m1 , K2=m2&m1=_ .
    [Show full text]
  • Hermite Polynomials 1 Hermite Polynomials
    Hermite polynomials 1 Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; in finite element methods as shape functions for beams; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator. They are also used in systems theory in connection with nonlinear operations on Gaussian noise. They were defined by Laplace (1810) [1] though in scarcely recognizable form, and studied in detail by Chebyshev (1859).[2] Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.[3] They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials. Definition There are two different ways of standardizing the Hermite polynomials: • The "probabilists' Hermite polynomials" are given by , • while the "physicists' Hermite polynomials" are given by . These two definitions are not exactly identical; each one is a rescaling of the other, These are Hermite polynomial sequences of different variances; see the material on variances below. The notation He and H is that used in the standard references Tom H. Koornwinder, Roderick S. C. Wong, and Roelof Koekoek et al. (2010) and Abramowitz & Stegun. The polynomials He are sometimes denoted by H , n n especially in probability theory, because is the probability density function for the normal distribution with expected value 0 and standard deviation 1.
    [Show full text]