28 May 2004 Umbral Calculus in Positive Characteristic
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Poly-Bernoulli Polynomials Arising from Umbral Calculus
Poly-Bernoulli polynomials arising from umbral calculus by Dae San Kim, Taekyun Kim and Sang-Hun Lee Abstract In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus. 1 Introduction The classical polylogarithmic function Lis(x) are ∞ xk Li (x)= , s ∈ Z, (see [3, 5]). (1) s ks k X=1 In [5], poly-Bernoulli polynomials are defined by the generating function to be −t ∞ n Li (1 − e ) (k) t k ext = eB (x)t = B(k)(x) , (see [3, 5]), (2) 1 − e−t n n! n=0 X (k) n (k) with the usual convention about replacing B (x) by Bn (x). As is well known, the Bernoulli polynomials of order r are defined by the arXiv:1306.6697v1 [math.NT] 28 Jun 2013 generating function to be t r ∞ tn ext = B(r)(x) , (see [7, 9]). (3) et − 1 n n! n=0 X (r) In the special case, r = 1, Bn (x)= Bn(x) is called the n-th ordinary Bernoulli (r) polynomial. Here we denote higher-order Bernoulli polynomials as Bn to avoid 1 conflict of notations. (k) (k) If x = 0, then Bn (0) = Bn is called the n-th poly-Bernoulli number. From (2), we note that n n n n B(k)(x)= B(k) xl = B(k)xn−l. (4) n l n−l l l l l X=0 X=0 Let F be the set of all formal power series in the variable t over C as follows: ∞ tk F = f(t)= ak ak ∈ C , (5) ( k! ) k=0 X P P∗ and let = C[x] and denote the vector space of all linear functionals on P. -
On Multivariable Cumulant Polynomial Sequences with Applications E
JOURNAL OF ALGEBRAIC STATISTICS Vol. 7, No. 1, 2016, 72-89 ISSN 1309-3452 { www.jalgstat.com On multivariable cumulant polynomial sequences with applications E. Di Nardo1,∗ 1 Department of Mathematics \G. Peano", Via Carlo Alberto 10, University of Turin, 10123, I-Turin, ITALY Abstract. A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment se- quences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L´evyprocesses and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper. 2000 Mathematics Subject Classifications: 62H05; 60C05; 11B83, 05E40 Key Words and Phrases: multi-index partition, cumulant, generating function, formal power series, L´evyprocess, exponential model 1. Introduction The so-called symbolic moment method has its roots in the classical umbral calculus developed by Rota and his collaborators since 1964; [15]. Rewritten in 1994 (see [17]), what we have called moment symbolic method consists in a calculus on unital number sequences. Its basic device is to represent a sequence f1; a1; a2;:::g with the sequence 2 f1; α; α ;:::g of powers of a symbol α; named umbra. The sequence f1; a1; a2;:::g is said to be umbrally represented by the umbra α: The main tools of the symbolic moment method are [7]: a) a polynomial ring C[A] with A = fα; β; γ; : : :g a set of symbols called umbrae; b) a unital operator E : C[A] ! C; called evaluation, such that i i) E[α ] = ai for non-negative integers i; ∗Corresponding author. -
Degenerate Poly-Bernoulli Polynomials with Umbral Calculus Viewpoint Dae San Kim1,Taekyunkim2*, Hyuck in Kwon2 and Toufik Mansour3
Kim et al. Journal of Inequalities and Applications (2015)2015:228 DOI 10.1186/s13660-015-0748-7 R E S E A R C H Open Access Degenerate poly-Bernoulli polynomials with umbral calculus viewpoint Dae San Kim1,TaekyunKim2*, Hyuck In Kwon2 and Toufik Mansour3 *Correspondence: [email protected] 2Department of Mathematics, Abstract Kwangwoon University, Seoul, 139-701, S. Korea In this paper, we consider the degenerate poly-Bernoulli polynomials. We present Full list of author information is several explicit formulas and recurrence relations for these polynomials. Also, we available at the end of the article establish a connection between our polynomials and several known families of polynomials. MSC: 05A19; 05A40; 11B83 Keywords: degenerate poly-Bernoulli polynomials; umbral calculus 1 Introduction The degenerate Bernoulli polynomials βn(λ, x)(λ =)wereintroducedbyCarlitz[ ]and rediscovered by Ustinov []underthenameKorobov polynomials of the second kind.They are given by the generating function t tn ( + λt)x/λ = β (λ, x) . ( + λt)/λ – n n! n≥ When x =,βn(λ)=βn(λ, ) are called the degenerate Bernoulli numbers (see []). We observe that limλ→ βn(λ, x)=Bn(x), where Bn(x)isthenth ordinary Bernoulli polynomial (see the references). (k) The poly-Bernoulli polynomials PBn (x)aredefinedby Li ( – e–t) tn k ext = PB(k)(x) , et – n n! n≥ n where Li (x)(k ∈ Z)istheclassicalpolylogarithm function given by Li (x)= x (see k k n≥ nk [–]). (k) For = λ ∈ C and k ∈ Z,thedegenerate poly-Bernoulli polynomials Pβn (λ, x)arede- fined by Kim and Kim to be Li ( – e–t) tn k ( + λt)x/λ = Pβ(k)(λ, x) (see []). -
L-Series and Hurwitz Zeta Functions Associated with the Universal Formal Group
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 133-144 L-series and Hurwitz zeta functions associated with the universal formal group PIERGIULIO TEMPESTA Abstract. The properties of the universal Bernoulli polynomials are illustrated and a new class of related L-functions is constructed. A generalization of the Riemann-Hurwitz zeta function is also proposed. Mathematics Subject Classification (2010): 11M41 (primary); 55N22 (sec- ondary). 1. Introduction The aim of this article is to establish a connection between the theory of formal groups on one side and a class of generalized Bernoulli polynomials and Dirichlet series on the other side. Some of the results of this paper were announced in the communication [26]. We will prove that the correspondence between the Bernoulli polynomials and the Riemann zeta function can be extended to a larger class of polynomials, by introducing the universal Bernoulli polynomials and the associated Dirichlet series. Also, in the same spirit, generalized Hurwitz zeta functions are defined. Let R be a commutative ring with identity, and R {x1, x2,...} be the ring of formal power series in x1, x2,...with coefficients in R.Werecall that a commuta- tive one-dimensional formal group law over R is a formal power series (x, y) ∈ R {x, y} such that 1) (x, 0) = (0, x) = x 2) ( (x, y) , z) = (x,(y, z)) . When (x, y) = (y, x), the formal group law is said to be commutative. The existence of an inverse formal series ϕ (x) ∈ R {x} such that (x,ϕ(x)) = 0 follows from the previous definition. -
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. -
A Note on Poly-Bernoulli Polynomials Arising from Umbral Calculus
Adv. Studies Theor. Phys., Vol. 7, 2013, no. 15, 731 - 744 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2013.3667 A Note on Poly-Bernoulli Polynomials Arising from Umbral Calculus Dae San Kim Department of Mathematics, Sogang University Seoul 121-742, Republic of Korea [email protected] Taekyun Kim Department of Mathematics, Kwangwoon University Seoul 139-701, Republic of Korea [email protected] Sang-Hun Lee Division of General Education Kwangwoon University Seoul 139-701, Republic of Korea [email protected] Copyright c 2013 Dae San Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we give some recurrence formula and new and interesting 732 Dae San Kim, Taekyun Kim and Sang-Hun Lee identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus. 1 Introduction The classical polylogarithmic function Lis(x) are ∞ k x ∈ Lis(x)= s ,s Z, (see [3, 5]). (1) k=1 k In [5], poly-Bernoulli polynomials are defined by the generating function to be −t ∞ n Lik (1 − e ) (k) t ext = eB (x)t = B(k)(x) , (see [3, 5]), (2) − −t n 1 e n=0 n! (k) n (k) with the usual convention about replacing B (x) by Bn (x). As is well known, the Bernoulli polynomials of order r are defined by the generating function to be r ∞ t tn ext = B(r)(x) , (see [7, 9]). -
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. -
Arxiv:Math/0402078V1 [Math.CO] 5 Feb 2004 Towards Ψ−Extension Of
Towards ψ−extension of Finite Operator Calculus of Rota A.K.Kwa´sniewski Institute of Computer Science, Bia lystok University PL-15-887 Bia lystok, ul.Sosnowa 64, POLAND Higher School of Mathematics and Applied Computer Science, PL-15-364 Bia lystok, Czysta 11 e-mail: [email protected] February 1, 2008 Abstract ψ- extension of Gian-Carlo Rota‘s finite operator calculus due to Viskov [1, 2] is further developed. The extension relies on the no- tion of ∂ψ-shift invariance and ∂ψ-delta operators. Main statements of Rota‘s finite operator calculus are given their ψ-counterparts. This includes Sheffer ψ-polynomials properties and Rodrigues formula - among others. Such ψ-extended calculus delivers an elementary um- bral underpinning for q-deformed quantum oscillator model and its possible generalisations. arXiv:math/0402078v1 [math.CO] 5 Feb 2004 ∂q-delta operators and their duals and similarly ∂ψ-delta operators with their duals are pairs of generators of ψ(q)- extended quantum n −1 oscillator algebras. With the choice ψn(q) = [R(q )!] and R(x) = 1−x 1−q we arrive at the well known q-deformed oscillator. Because the reduced incidence algebra R(L(S)) is isomorphic to the algebra Φψ of ψ -exponential formal power series - the ψ-extensions of finite operator calculus provide a vast family of representations of R(L(S)). KEY WORDS: extended umbral calculus, quantum q-plane MSC(2000): 05A40, 81S99 1 1 Introduction The main aim of this paper is a presentation of ψ- extension of Rota‘s finite operator calculus simultaneously with an indication that this is a natural and elementary method for formulation and treatment of q-extended and posibly R- extended or ψ-extended models for quantum-like ψ-deformed oscillators in order to describe [3] eventual processes with parastatistical behavior. -
Extended Finite Operator Calculus
CEJM 2(5) 2005 767–792 Review article Extended finite operator calculus - an example of algebraization of analysis Andrzej Krzysztof Kwa´sniewski1∗, Ewa Borak2 1 Higher School of Mathematics and Applied Informatics, 15-021 Bialystok , ul.Kamienna 17, Poland 2 Institute of Computer Science, Bia lystok University, 15-887 Bia lystok, ul.Sosnowa 64, Poland Received 15 December 2003; accepted 21 October 2004 Abstract: “A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota - Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward‘s calculus of sequences in its afterwards elaborated form - a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis - here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8] , Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]). -
An Infinite Dimensional Umbral Calculus
An infinite dimensional umbral calculus Dmitri Finkelshtein Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.; e-mail: [email protected] Yuri Kondratiev Fakult¨atf¨urMathematik, Universit¨atBielefeld, 33615 Bielefeld, Germany; e-mail: [email protected] Eugene Lytvynov Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.; e-mail: [email protected] Maria Jo~aoOliveira Departamento de Ci^enciase Tecnologia, Universidade Aberta, 1269-001 Lisbon, Por- tugal; CMAF-CIO, University of Lisbon, 1749-016 Lisbon, Portugal; e-mail: [email protected] Abstract The aim of this paper is to develop foundations of umbral calculus on the space D0 of distri- d 0 butions on R , which leads to a general theory of Sheffer polynomial sequences on D . We define a sequence of monic polynomials on D0, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D0 to be of binomial type or a Sheffer sequence, respectively. We also construct a lift- ing of a sequence of monic polynomials on R of binomial type to a polynomial sequence of 0 0 binomial type on D , and a lifting of a Sheffer sequence on R to a Sheffer sequence on D . Examples of lifted polynomial sequences include the falling and rising factorials on D0, Abel, Hermite, Charlier, and Laguerre polynomials on D0. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role. arXiv:1701.04326v2 [math.FA] 5 Mar 2018 Keywords: Generating function; polynomial sequence on D0; polynomial sequence of binomial type on D0; Sheffer sequence on D0; shift-invariance; umbral calculus on D0. -
Differential Operators and the Theory of Binomial Enumeration
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 63, 145-155 (1978) Differential Operators and the Theory of Binomial Enumeration GEORGE MARKOWSKY Computer Science Deportment, IBM Thomas J. Watson Research Center, Yorktomtt Heights, New York 10598 Submitted by G.-C. Rota I. INTRODUCTION In [4], Rota and Mullin established a very interesting correspondence between polynomials of “binomial type” and a family of linear operators they call “delta operators.” In this paper we consider the family of all differential operators (linear operators which lower degree by one) on the ring of polynomials over a field, F, of characteristic 0. We show that this family breaks up into subfamilies based on commutativity. We show many of the results in [4], generalize to these subfamilies, and that these results depend upon straightforward manipulations of formal power series and the use of Taylor’s Theorem, rather than on specific substitution-type arguments. These methods allow us to show that in order for an operator to be shift-invariant (i.e., to commute with all shift-operators) it is enough for it to commute with a single nontriwiul (#I) shift operator. The key result here is that given two differential operators G and H, G can be expanded in terms of H if and only if GH = HG. We derive the operator form of the Pincherle Derivative (the formal power series derivative) and use this result to extend the closed form formulas of Rota and Mullin to the “basic families” of polynomials of arbitrary differential operators. -
Analysis of Bell Based Euler Polynomials and Thier Application
ANALYSIS OF BELL BASED EULER POLYNOMIAL AND THEIR APPLICATION NABIULLAH KHAN AND SADDAM HUSAIN Abstract. In the present article, we study Bell based Euler polynomial of order α and investigate some useful correlation formula, summation formula and derivative formula . Also, we introduce some relation of string number of the second kind. Moreover, we drive several important formulae of bell based Euler polynomial by using umbral calculus. 1. Introduction The polynomials and number play an important role in the multifarious area of science such as mathematics, applied science, physics and engineering sciences and some related research area involving number theory, quantum mechanics, dif- ferential equation and mathematical physics (see [1, 3, 4]). Some of the most important polynomials are Bell polynomial, Genocchi polynomial, Euler polyno- mial, Bernaulii polynomial and Hermite polynomial. We know that, properties of polynomials with the help of umbral calculus studied by many authors. Kim et al. [11] study some properties of Euler polynomials by using umbral Calculus and give some useful identities of Euler polynomials. Kim et al. [12] study Bernoulli, Euler and Abel polynomials by using umbral calculus and gives interesting identities of it. Kim et al. [14] investigate useful identities for partially degenerate Bell Number and polynomials associated with umbral calculus and Kim et al. [15] drived some important identities and properties of umbral calculus associated with degenerate orderd Bell number and polynomials. arXiv:2104.09129v1 [math.NT] 19 Apr 2021 Recently, S. Araci et al. [7] defined a Bell based Bernaoulli polynomials. Moti- vated by above mention work in the present article we introduce Bell based Euler polynomials and investigate some useful correlation formula, summation formula and derivative formula of Bell based Euler polynomial of order α.