DOCSLIB.ORG

Self-Inverse Sheffer Sequences and Riordan Involutions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Self-Inverse Sheffer Sequences and Riordan Involutions Discrete Applied Mathematics 159 (2011) 1290–1292 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note Self-inverse Sheffer sequences and Riordan involutions Ana Luzón a,∗, Manuel A. Morón b a Departamento de Matemática Aplicada a los Recursos Naturales, E.T. Superior de Ingenieros de Montes, Universidad Politécnica de Madrid, 28040-Madrid, Spain b Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense de Madrid, 28040- Madrid, Spain article info a b s t r a c t Article history: In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan Received 23 October 2009 group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer Received in revised form 7 December 2010 polynomials to describe all involutions in the Riordan group. Accepted 25 March 2011 ' 2011 Elsevier B.V. All rights reserved. Available online 8 May 2011 Keywords: Riordan group Involution Self-inverse Sheffer sequence Very recently, first in [7] and later in [17], see also [12], a very close relation has been established between the Sheffer group and the Riordan group; see [15,16]. In fact, there is a natural isomorphism between both groups. This means that the group properties can be translated from one group to the other equivalently. One of those properties is the structure of their finite subgroups. In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan group. They determine the corresponding subgroups of order 2. In fact, we translate the results of Brown and Kuczma [2] on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group. Although the translation is almost automatic, we think that it is still interesting to point it out, because of the relations to some problems about involutions in the Riordan group posed in [14] that motivated the paper [3] and that have recently been solved in [5]; see also [6,4]. In [8], the authors give some properties of the so-called A and Z sequences of any Riordan involution. There are also some related results on pages 2264–2265 of [11]. In a sense, we want to point out that some aspects of Shapiro's problem were solved, even before it was posed, if we reinterpret it in terms of Sheffer sequences. In this paper, K always represents a field of characteristic zero and N is the set of natural numbers including 0. The notation used herein for Riordan arrays is that introduced in [10]; see also [11]. The natural translation from our notation to the classical notation is given by the formula f .t/ t T .f jg/ D ; (1) g.t/ g.t/ with g.0/ 6D 0. Up to the inconvenience that produces the fact that we call, following [1], a generalized Appell sequence associated D P n to Hadamard invertible series h.x/ n≥0 hnx simply the sequence obtained by multiplying by hn the n-term of the polynomial sequence named in the same way in [2], we have the following obvious result. The notation used below is that ∗ Corresponding author. Tel.: +34 913 366 399; fax: +34 915 439 557. E-mail addresses: [email protected] (A. Luzón), [email protected] (M.A. Morón). 0166-218X/$ – see front matter ' 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2011.03.025 A. Luzón, M.A. Morón / Discrete Applied Mathematics 159 (2011) 1290–1292 1291 x used in [12], the operation group ]h is the umbral composition as is also described in [7] for the particular case h.x/ D e , and ? represents the Hadamard product of series. D P n Proposition 1. Let N be a natural number. Suppose a polynomial sequence of Riordan type .pn.x//n2N and let h.x/ n≥0 hnx 6D 8 2 h D be a series with hn 0 n N. Consider the Hadamard h-weighted sequence .pn.x// .pn.x/? h.x//. Then, the N-fold umbral h N D composition, by means of ]h, of the sequence .pn.x// is the neutral element in .Rh;]h/ if and only if D I in the Riordan group, D D Pn k where D .dn;k/ is the Riordan matrix given by pn.x/ kD0 dn;kx . n Remark 2. Note that the sequence en.x/ D hnx is the neutral element in .Rh;]h/. We now translate the results in [2] on self-inverse Sheffer sequences into Riordan involutions. In particular, using the results in Section 3 of [2], we can give a procedure to compute all the elements T .f j g/ of the Riordan group such that T 2.f j g/ D T .1 j 1/ D I. First, if we impose g D 1, then T 2.f j g/ D T .1 j 1/ if and only if f D 1 or f D −1. To construct the remaining cases we proceed in the following way. D P n 6D j Choose any series φ n≥0 φnx with φ0 0. Consider the Riordan matrix T .1 φ/. As a consequence of the results in [9], we get that T .1 j A/ D T −1.1 j φ/, where A is the so-called A-sequence of T .1 j φ/; see [16,13]. It is clear that x x D x x D x, that is, x is the inverse for the composition of the series x . Following [2], we take A φ φ A A φ x g D : (2) x − x A φ D P 2k−1 Choose now any odd power series u k≥1 u2k−1x , and take ±x x u φ f D e : (3) x − x A φ Consequently, we have the following. Proposition 3. Any Riordan involution different from the identity I D T .1 j 1/ and from −I D T .−1 j 1/ is of the form T .f j g/ for f and g satisfying (3) and (2), respectively. We want to point out that the pair of series f and g above is far from being univocally determined by φ and u. For example, consider the following. Proposition 4. Let D P xn be a series with 6D 0. Suppose that x is the compositional inverse of x (equivalently that φ n≥0 φn φ0 A φ x A is the A-sequence of T .1 j φ/). Suppose also that g D . Then g D −1 if and only if φ.x/ D φ.−x/ (i.e., φ is even). x − x A φ D − x Proof. Note that φ.x/ φ. x/ if and only if the series φ is odd. If is even, then x is odd. Since x is the compositional inverse of an odd power series, x is odd itself. Consequently, φ φ A A − g D −1. On the other hand, if g D −1, then x − x D −x. Composing the right-hand side by x , we have x −x D x D A φ A A . / A.−x/ − x . This implies that x is odd, and then is even. A.x/ φ φ As we proved on page 2265 of [11], the Riordan matrices T .±1 j αx − 1/ are involutions for any α 6D 0. Now, we are going to recover this result using the construction above. In fact, we will get a more general class of Riordan involutions. Let α 6D 0, and take αx φ.x/ D : log.1 − αx/ Consequently, αx A.x/ D : 1 − eαx This implies that x g D D αx − 1: x − x A φ We know that, if u 1 log.1−αx/ f D ±(αx − 1/e α ; 1292 A. Luzón, M.A. Morón / Discrete Applied Mathematics 159 (2011) 1290–1292 then T .f j g/ is an involution when u is an odd series. In particular, if u.x/ D −αx, then we obtain that T .±1 j αx − 1/ is a Riordan involution. From this point of view we have the following. The fact that the Pascal triangle is a pseudo-involution (see [3]) is equivalent to the fact that the classical family of Laguerre polynomials is self-inverse (see [2]). Remark 5. Theorem 4.3 in [8] gives the expressions of the A-sequence and the Z-sequence of any Riordan involution. Using this theorem and the description of the involutions given in Proposition 3 above, we get that, if T .fQjgQ/ is an involution and AQ u x Q x Q ±xe φ Q x and Z are the corresponding A-sequence and Z-sequence, respectively, then gQ.x/ D , f .x/ D , A.x/ D x − x x − x x − x A φ A φ A φ u x Q δ±e φ and Z.x/ D , where φ is a series with φ.0/ 6D 0, A is the A-sequence of T .1jφ/, and u is any odd power series. So the x − x A φ u x non-trivial Riordan involutions in the classical notation are ±e φ x − x , or equivalently ±eu.!.x// −1 − x , ; A φ . ;! . !. /// where ! is any power series invertible for the composition operation. Moreover, using Theorem 4.3 in [8], we obtain − u.!.x// AQ.x/ D x and ZQ.x/ D ± 1 e . !−1.−!.x// !−1.−!.x// The results in Theorem 4.3 in [8] are also reflected in the formula T −1.f j g/ D T g0 .A − xZ/ A , which appears f0 in [9].
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • 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]
  • 2-Iterated Sheffer Polynomials
    2-iterated Sheffer polynomials⋆ Subuhi Khan* and Mumtaz Riyasat Department of Mathematics Aligarh Muslim University Aligarh, India Abstract: In this article, the 2-iterated Sheffer polynomials are introduced by means of gener- ating function and operational representation. Using the theory of Riordan arrays and relations between the Sheffer sequences and Riordan arrays, a determinantal definition for these polyno- mials is established. The quasi-monomial and other properties of these polynomials are derived. The generating function, determinantal definition, quasi-monomial and other properties for some new members belonging to this family are also considered. Keywords: 2-iterated Sheffer polynomials; Differential equation; Determinantal definition. 1. Introduction and preliminaries Sequences of polynomials play a fundamental role in mathematics. One of the most famous classes of polynomial sequences is the class of Sheffer sequences, which contains many important sequences such as those formed by Bernoulli polynomials, Euler polynomials, Abel polynomi- als, Hermite polynomials, Laguerre polynomials, etc. and contains the classes of associated sequences and Appell sequences as two subclasses. Roman et. al. in [27,28] studied the Sheffer sequences systematically by the theory of modern umbral calculus. Roman [26] further developed the theory of umbral calculus and generalized the concept of Sheffer sequences so that more special polynomial sequences are included, such as the sequences related to Gegenbauer polynomials, Chebyshev polynomials and Jacobi polynomials. Let K be a field of characteristic zero. Let F be the set of all formal power series in the variable t over K. Thus an element of F has the form ∞ k arXiv:1506.00087v1 [math.CA] 30 May 2015 f(t)= akt , (1.1) Xk=0 where ak ∈ K for all k ∈ N := {0, 1, 2,...}.
    [Show full text]
  • Recurrence Relations for the Sheffer Sequences
    Sheffer sequences Exponential Riordan array Recurrent relations Examples Recurrence relations for the Sheffer sequences Sheng-liang Yang Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, P.R. China E-mail address: [email protected] 2012 Shanghai Conference on Algebraic Combinatorics August 17-22, 2012 Shanghai Jiao Tong University tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples Outline 1 Sheffer sequences 2 Exponential Riordan array 3 Recurrent relations 4 Examples tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples Introduction In this talk, using the production matrix of an exponential Ri- ordan array [g(t); f (t)], we give a recurrence relation for the Shef- fer sequence for the ordered pair (g(t); f (t)) . We also develop a new determinant representation for the general term of the Sheffer sequence. As applications, determinant expressions for some classical Sheffer polynomial sequences are derived. tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 1. Sheffer sequences polynomial sequence: p0(x), p1(x), p2(x), ··· , pk(x), ··· . deg pk(x) = k, for all k ≥ 0. Sheffer sequence: s0(x), s1(x), s2(x), ··· , sk(x), ··· . Let g(t) be an invertible series and let f (t) be a delta series; we say that the polynomial sequence sn(x) is the Sheffer sequence for the pair (g(t); f (t)) if and only if 1 n X t 1 xf (t) sn(x) = e : (1) n! g(f (t)) n=0 where f (t) is the compositional inverse of f (t). tu-logo Sheffer sequences Exponential Riordan array Recurrent relations Examples 1.
    [Show full text]
  • 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.
    [Show full text]
  • A Note on Appell Sequences, Mellin Transforms and Fourier Series
    A NOTE ON APPELL SEQUENCES, MELLIN TRANSFORMS AND FOURIER SERIES LUIS M. NAVAS, FRANCISCO J. RUIZ, AND JUAN L. VARONA 1 Abstract. A large class of Appell polynomial sequences fpn(x)gn=0 are special values at the negative integers of an entire function F (s; x), given by the Mellin transform of the generating function for the se- quence. For the Bernoulli and Apostol-Bernoulli polynomials, these are basically the Hurwitz zeta function and the Lerch transcendent. Each of these have well-known Fourier series which are proved in the litera- ture using varied techniques. Here we find the latter Fourier series by directly calculating the coefficients in a straightforward manner. We then show that, within the context of Appell sequences, these are the only cases for which the polynomials have uniformly convergent Fourier series. In the more general context of Sheffer sequences, we find that there are other polynomials with uniformly convergent Fourier series. Finally, applying the same ideas to the Fourier transform, considered as the continuous analog of the Fourier series, the Hermite polynomials play a role analogous to that of the Bernoulli polynomials. 1. Introduction 1 An Appell sequence fPn(x)gn=0 is defined formally by an exponential generating function of the form 1 X tn (1) G(x; t) = A(t) ext = P (x) ; n n! n=0 where x; t are indeterminates and A(t) is a formal power series. One often adds the condition A(0) 6= 0, although in principle this is not necessary. As in [11], we define an Appell-Mellin sequence as an Appell sequence where A(t) is a function defined on the union of a complex neighborhood of the origin and the negative axis (−∞; 0), satisfying: (a) A(t) is non-constant and analytic around 0, (b) A(−t) is continuous on [0; +1) and has polynomial growth at +1.
    [Show full text]
  • Sheffer Sequences, Probability Distributions and Approximation
    She®er sequences, probability distributions and approximation operators M. Cr¸aciun a;1 and A. Di Bucchianico b;¤ a\T. Popoviciu" Institute of Numerical Analysis, 3400 Cluj{Napoca, Romania bEindhoven University of Technology, Department of Mathematics, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Abstract We present a new method to compute formulas for the action on monomials of a generalization of binomial approximation operators of Popoviciu type, or equiva- lently moments of associated discrete probability distributions with ¯nite support. These quantities are necessary to check the assumptions of the Korovkin Theorem for approximation operators, or equivalently the Feller Theorem for convergence of the probability distributions. Our method uni¯es and simpli¯es computations of well-known special cases. It only requires a few basic facts from Umbral Calculus. We illustrate our method to well-known approximation operators and probability distributions, as well as to some recent q-generalizations of the Bernstein approxi- mation operator introduced by Lewanowicz and Wo¶zny, Lupa»s,and Phillips. Key words: approximation operators of Popoviciu type, moments, Umbral Calculus, She®er sequences 1 Introduction A generalization of binomial approximation operators of Popoviciu type was studied in [1]. Many well-known linear positive approximation operators be- long to this class, like the Bernstein, Stancu and Cheney-Sharma operators. ¤ Corresponding author. Email addresses: [email protected] (M. Cr¸aciun), [email protected] (A. Di Bucchianico). URL: www.win.tue.nl/»sandro (A. Di Bucchianico). 1 Supported by grants 27418/2000, GAR 15/2003, GAR/13/2004 and visitor grants from Eindhoven University of Technology, all of which are gratefully acknowledged.
    [Show full text]
  • First Contact Remarks on Umbra Difference Calculus References
    First Contact Remarks on Umbra Difference Calculus References Streams A.K.Kwa´sniewski Higher School of Mathematics and Applied Informatics PL-15-021 Bia lystok,ul. Kamienna 17, POLAND e-mail: [email protected] April 18, 2004 ArXiv: math.CO/0403139 v1 8 March 2004 Abstract The reference links to the modern "classical umbral calculus" (be- fore that properly called Blissard`s symbolic method) and to Steffensen -actuarialist.....are numerous. The reference links to the EFOC (Ex- tended Finite Operator Calculus) founded by Rota with numerous out- standing Coworkers, Followers and Others ... and links to Roman-Rota functional formulation of umbra calculi ... .....are giant numerous. The reference links to the difference q-calculus - umbra-way treated or with- out even referring to umbra.... .....are plenty numerous. These refer- ence links now result in counting in thousands the relevant papers and many books . The purpose of the present attempt is to offer one of the keys to enter the world of those thousands of references. This is the first glimpse - not much structured - if at all. The place of entrance was chosen selfish being subordinated to my present interests and workshop purposes. You are welcomed to add your own information. Motto Herman Weyl April 1939 The modern evolution has on the whole been marked by a trend of algebraization ... - from an invited address before AMS in conjunction with the cetential celebration of Duke University. 1 I. First Contact Umbral Remark "R-calculus" and specifically q -calculus - might be considered as specific cases of umbral calculus as illustrated by a] and b] below : 1 a] Example 2.2.from [1] quotation:...."The example of -derivative: Q(@ ) = R(qQ^)@0 @R ≡ i.e.
    [Show 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].
    [Show full text]
  • A Series Representation for the Riemann Zeta Derived from the Gauss-Kuzmin-Wirsing Operator
    A series representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator Linas Vepstas <[email protected]> 2 January 2004 (revised 29 August 2005) Abstract A series representation for the Riemann zeta function in terms of the falling Pochhammer symbol is derived from the polynomial representation of the Gauss- Kuzmin-Wirsing (GKW) operator. 1 Introduction The Gauss-Kuzmin-Wirsing operator [Kuz28, Wir74, Ba78a] occurs in the theory of continued fraction representations of the real numbers. The zeroth eigenvalue of this operator is related to the Gauss-Kuzmin distribution [ref], giving the likelihood of the occurrence of an integer in the continued-fractionexpansionof an “arbitrary”real num- ber. The GKW operator is interesting in other ways, notably in its relationship to the Minkowski Question Mark function, and thus its symmetry under the action of a monoid sub-semi-group of the modular group SL(2,Z). Except for the zeroth eigen- vector, there is no known closed-form solution of the GKW operator, although the eigenvalues may be computed relatively easily through standard matrix diagonaliza- tion techniques. In this paper, a relationship to the Riemann zeta function [Ed74] is noted, allowing the easy derivation of a series expansion of the zeta function in terms of the Pochham- mer symbols (the falling factorials), or equivalently the binomial coefficients. Specifi- cally, the expansion given is ∞ s n s 1 ζ(s)= s ∑ ( ) − tn (1) s 1 − − n − n=0 Here, the constants tn play a role analogous to the Stieltjes constants, and can be given in a simple, finite-term closed form, involving the Riemann zeta function at integer values and the Euler-Mascheroni constant.
    [Show full text]