DOCSLIB.ORG

A RESEARCH on the GENERALIZED POLY-BERNOULLI POLYNOMIALS with VARIABLE A†

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A RESEARCH on the GENERALIZED POLY-BERNOULLI POLYNOMIALS with VARIABLE A† J. Appl. Math. & Informatics Vol. 36(2018), No. 5 - 6, pp. 475 - 489 https://doi.org/10.14317/jami.2018.475 A RESEARCH ON THE GENERALIZED POLY-BERNOULLI POLYNOMIALS WITH VARIABLE ay N.S. JUNG AND C.S. RYOO∗ Abstract. In this paper, by using the polylogarithm function, we intro- duce a generalized poly-Bernoulli numbers and polynomials with variable a. We find several combinatorial identities and properties of the polynomi- als. We give some properties that is connected with the Stirling numbers of second kind. Symmetric properties can be proved by new configured special functions. We display the zeros of the generalized poly-Bernoulli polynomials with variable a and investigate their structure. AMS Mathematics Subject Classification : 11B68, 11B73, 11B75. Key words and phrases : Generalized poly-Bernoulli polynomials with variable a, Stirling numbers of the second kind, polylogarithm function, power sum polynomials, special function. 1. Introduction Recently, one of the researchers' interests in many fields is the applications of Bernoulli numbers and polynomials. Many researchers have studied Bernoulli numbers and polynomials and focus on expansion and generalization of theirs. Specially, it is being studied actively about poly-Bernoulli numbers and polyno- mials is concerned with polylogarithm function(cf. [1-9]). In this paper, we use the following notations. N = f1; 2; 3;::: g denotes the set of natural numbers, Z+ denotes the set of nonnegative integers, Z denotes the set of integers, and C denotes the set of complex numbers, respectively. The classical Bernoulli polynomials Bn(x) are given by the generating function 1 t X tn ext = B (x) ; (cf. [1, 2, 3, 4, 5]): (1:1) et − 1 n n! n=0 (k) (k) When x = 0;Bn;q = Bn;q(0) are called poly-Bernoulli numbers. Received January 9, 2018. Revised July 25, 2018. Accepted August 5, 2018. ∗Corresponding author. yThis work was supported by 2018 Hannam University Research Fund. ⃝c 2018 Korean SIGCAM and KSCAM. 475 476 N.S. Jung, C.S. Ryoo Definition 1.1. For a 2 C n f0g, we define a generalized Bernoulli polynomials Bn(x; a) with variable a by the following generating function 1 t X tn 2π ext = B (x; a) ; jtj < : (1:2) eat − 1 n n! jaj n=0 When a = 1, it is equal to the classical Bernoulli polynomials. The polyloga- rithm function Lik(x) is defined by 1 X xn Li (x) = ; (k 2 Z); (cf. [1, 2, 3, 5, 7, 8, 9, 12]): (1:3) k nk n=1 For some k, the polylogarithm functions Lik(x) are as follows: x Li (x) = −log(1 − x); Li (x) = ; 1 0 1 − x x x2 + x Li−1(x) = ; Li−2(x) = ;:::: (1 − x)2 (1 − x)3 For n 2 Z+; k 2 Z, the poly-Bernoulli polynomials is defined by 1 Li (1 − e−t) X tn k ext = B(k)(x) ; (1:4) et − 1 n n! n=0 where 1 X xn Li (x) = k nk n=1 is k-th polylogarithm function(cf. [1, 2, 3, 4, 5, 7, 12]). When k = 1, Li1(x) = −t −log(1−x) and Li1(1−e ) = t. Using the result of polylogarithm function, we deduce that the poly Bernoulli polynomials is identical to the Bernoulli polyno- mials when k = 1. The classical Stirling numbers of the second kind S2(n; m) are defined by the relations Xn n x = S2(n; m)(x)m; m=0 where (x)n = x(x − 1)(x − 2) ··· (x − n + 1) is falling factorial. The Stirling numbers of the second kind is defined by 1 X tn (et − 1)m S (n; m) = ; (cf. [4, 6, 9, 10, 11]): (1:5) 2 n! m! n=m In this paper, we introduce a generalized poly-Bernoulli numbers and polynomi- als with variable a. The properties of the Bernoulli polynomials with parameters were studied in [5, 8]. We construct a generalized poly-Bernoulli polynomials with variable a and give some relations between the generalized poly-Bernoulli polynomials and the classical Bernoulli polynomials. We also investigate sev- eral identities that are connected with the Stirling numbers of the second kind. A research on the generalized poly-Bernoulli polynomials with variable a 477 Furthermore, we find some symmetric identities by using special functions and power sum polynomials. 2. Generalized poly-Bernoulli polynomials with variable a (k) In this section, we introduce a generalized poly-Bernoulli numbers Bn (a) (k) and polynomials Bn (x; a) with variable a by the generating functions. We give some identities of the polynomials, and find a relation that is connected with classical Bernoulli polynomials. Definition 2.1. For n 2 Z+ and k 2 Z, the generalized poly-Bernoulli polyno- (k) mials Bn (x; a) with variable a are defined by means of the following generating function 1 Li (1 − e−t) X tn k ext = B(k)(x; a) (2:1) eat − 1 n n! n=0 where 1 X tn Li (t) = k nk n=1 (k) (k) is the k-th polylogarithm function. When x = 0, Bn (a) = Bn (0; a) are called the generalized poly-Bernoulli numbers with variable a. When the condition al- low a = 1, it is trivial that the generalized poly-Bernoulli polynomials is reduced to poly-Bernoulli polynomials. From (2.1), we have a relation between the generalized poly-Bernoulli numbers and polynomials. Theorem 2.2. Let n; m be a nonnegative integers and k 2 Z. We have ( ) Xn n B(k)(mx; a) = (m − 1)n−1B(k)(x; a)xn−l: (2:2) n l l l=0 Proof. For n; m 2 Z+, k 2 Z; we get ! 1 1 ( ) X tn X Xn n tn B(k)(mx; a) = B(k)(x; a)(m − 1)n−1xn−l : n n! l l n! n=0 n=0 l=0 Therefore, we obtain above result. Corollary 2.3. Let m > 0, n ≥ 0, k 2 Z. We get ( ) Xn n B(k)(mx; a) = mn−lB(k)(a)xn−l: n l l l=0 When m = 1, it is satisfies ( ) Xn n B(k)(x; a) = B(k)(a)xn−l: n l l l=0 478 N.S. Jung, C.S. Ryoo If x is replaced x + y in Corollary 2.3, we get the next addition theorem. Theorem 2.4. For n 2 Z+ and k 2 Z, we have ( ) Xn n B(k)(x + y; a) = B(k)(x; a)yn−l: n l l l=0 Proof. Let n 2 Z+, k 2 Z. Then we get 1 X tn Li (1 − e−t) B(k)(x + y; a) = k e(x+y)t n n! eat − 1 n=0 ! 1 ( ) X Xn n tn = B(k)(x; a)yn−l : l l n! n=0 l=0 Thus, we get the explicit result. Theorem 2.5. For n 2 Z+ and k 2 Z, we derive − ( ) nX1 n B(k)(x + 1; a) − B(k)(x; a) = B(k)(x; a): n n l l l=0 Proof. Let n 2 Z+; k 2 Z. From (2.1), we have 1 1 X tn X tn B(k)(x + 1; a) − B(k)(x; a) n n! n n! n=0 n=0 Li (1 − e−t) = k ext(et − 1) eat − 1 1 − ( ) X nX1 n tn = B(k)(x; a) : l l n! n=0 l=0 Comparing the coefficient on both sides, we obtain the desired result. By using the binomials series and the definition of polylogarithm function, we derive the result as below. Theorem 2.6. For n 2 Z+ and k 2 Z, we have 1 ( ) X Xl mX+1 m + 1 (−1)r+1(x − r + al − am)n B(k)(x; a) = : n r (m + 1)n l=0 m=0 r=0 A research on the generalized poly-Bernoulli polynomials with variable a 479 Proof. Let n 2 Z+; k 2 Z: From (1.3), we obtain ! ! 1 1 Li (1 − e−t) X X (1 − e−t)l+1 k ext = (−1) emat ext eat − 1 (l + 1)k m=0 l=0 1 X Xl (1 − e−t)m+1 = (−1) e(l−m)at ext (m + 1)k l=0 m=0 ! ! 1 ( ) X Xl e(l−m)at mX+1 m + 1 = (−1) (−1)re(x−r)t (m + 1)k r l=0 m=0 r=0 1 1 ( ) X X Xl mX+1 m + 1 (−1)r+1(x − r + al − am)n tn = r (m + 1)k n! n=0 l=0 m=0 r=0 Therefore, we get 1 ( ) X Xl mX+1 m + 1 (−1)r+1(x − r + al − am)n B(k)(x; a) = : n r (m + 1)k l=0 m=0 r=0 Similarly, we find next result that is related with the generalized Bernoulli polynomials with variable a. Theorem 2.7. Let n 2 Z+ and k 2 Z: Then we have 1 ( ) X 1 Xl+1 l + 1 B (x − r; a) B(k)(x; a) = (−1)r n+1 : n (l + 1)k r n + 1 l=0 r=0 Proof. For n 2 Z+ and k 2 Z, we have 1 Li (1 − e−t) X (1 − e−t)l ext k ext = eat − 1 lk eat − 1 l=1 1 ( ) X 1 Xl+1 l + 1 e(x−r)t = (−1)r (l + 1)k r eat − 1 n=0 r=0 ! 1 1 ( ) X X 1 Xl+1 l + 1 B (x − r; a) tn = (−1)r n+1 : (l + 1)k r n + 1 n! n=0 l=0 r=0 So, we easily get 1 ( ) X 1 Xl+1 l + 1 B (x − r; a) B(k)(x; a) = (−1)r n+1 : n (l + 1)k r n + 1 l=0 r=0 480 N.S.
Load more

Recommended publications
  • An Identity for Generalized Bernoulli Polynomials
    1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.11.2 47 6 23 11 An Identity for Generalized Bernoulli Polynomials Redha Chellal1 and Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria [email protected] [email protected] [email protected] Mohamed Mehbali Centre for Research Informed Teaching London South Bank University London United Kingdom [email protected] Abstract Recognizing the great importance of Bernoulli numbers and Bernoulli polynomials in various branches of mathematics, the present paper develops two results dealing with these objects. The first one proposes an identity for the generalized Bernoulli poly- nomials, which leads to further generalizations for several relations involving classical Bernoulli numbers and Bernoulli polynomials. In particular, it generalizes a recent identity suggested by Gessel. The second result allows the deduction of similar identi- ties for Fibonacci, Lucas, and Chebyshev polynomials, as well as for generalized Euler polynomials, Genocchi polynomials, and generalized numbers of Stirling. 1Corresponding author. 1 1 Introduction Let N and C denote, respectively, the set of positive integers and the set of complex numbers. (α) In his book, Roman [41, p. 93] defined generalized Bernoulli polynomials Bn (x) as follows: for all n ∈ N and α ∈ C, we have ∞ tn t α B(α)(x) = etx. (1) n n! et − 1 Xn=0 The Bernoulli numbers Bn, classical Bernoulli polynomials Bn(x), and generalized Bernoulli (α) numbers Bn are, respectively, defined by (1) (α) (α) Bn = Bn(0), Bn(x)= Bn (x), and Bn = Bn (0). (2) The Bernoulli numbers and the Bernoulli polynomials play a fundamental role in various branches of mathematics, such as combinatorics, number theory, mathematical analysis, and topology.
    [Show full text]
  • A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function
    mathematics Article A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function Daeyeoul Kim 1,* and Yilmaz Simsek 2 1 Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Korea 2 Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya TR-07058, Turkey; [email protected] * Correspondence: [email protected] Abstract: In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials. Keywords: Bernoulli numbers and polynomials; Euler numbers and polynomials; Apostol–Bernoulli and Apostol–Euler numbers and polynomials; Hurwitz–Lerch zeta function; Hurwitz zeta function; alternating Hurwitz zeta function; generating function; Mellin transformation MSC: 05A15; 11B68; 26C0; 11M35 Citation: Kim, D.; Simsek, Y. A New Family of Zeta Type Function 1. Introduction Involving the Hurwitz Zeta Function The families of zeta functions and special numbers and polynomials have been studied and the Alternating Hurwitz Zeta widely in many areas. They have also been used to model real-world problems.
    [Show full text]
  • The Euler-Maclaurin Summation Formula and Bernoulli Polynomials R 1 Consider Two Different Ways of Applying Integration by Parts to 0 F(X)Dx
    The Euler-Maclaurin Summation Formula and Bernoulli Polynomials R 1 Consider two different ways of applying integration by parts to 0 f(x)dx. Z 1 Z 1 Z 1 1 0 0 f(x)dx = [xf(x)]0 − xf (x)dx = f(1) − xf (x)dx (1) 0 0 0 Z 1 Z 1 Z 1 1 0 f(1) + f(0) 0 f(x)dx = [(x − 1/2)f(x)]0 − (x − 1/2)f (x)dx = − (x − 1/2)f (x)dx (2) 0 0 2 0 The constant of integration was chosen to be 0 in (1), but −1/2 in (2). Which way is better? In general R 1 (2) is better. Since 0 (x − 1/2)dx = 0, that is x − 1/2 has zero average in (0, 1), it has better chance of reducing the last integral term, which is regarded as the error in estimating the given integral. We can again apply integration by parts to (2). Then the last integral will have f 00(x) multiplied by an integral of x − 1/2. We can select the constant of integration in such a way that the new polynomial multiplying f 00(x) will have zero average in the interval. If we continue this way we will get a formula which will have a higher derivative of f in the last integral multiplying a higher degree polynomial in x. If at each step the constant of integration is chosen in such a way that the new polynomial has zero average over (0, 1), we will get the Euler-Maclaurin summation formula.
    [Show full text]
  • Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
    Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis
    [Show full text]
  • Euler-Maclaurin Summation Formula
    4 Euler-Maclaurin Summation Formula 4.1 Bernoulli Number & Bernoulli Polynomial 4.1.1 Definition of Bernoulli Number Bernoulli numbers Bk ()k =1,2,3, are defined as coefficients of the following equation. x Bk k x =Σ x e -1 k=0 k! 4.1.2 Expreesion of Bernoulli Numbers B B x 0 1 B2 2 B3 3 B4 4 = + x + x + x + x + (1,1) ex-1 0! 1! 2! 3! 4! ex-1 1 x x2 x3 x4 = + + + + + (1.2) x 1! 2! 3! 4! 5! Making the Cauchy product of (1.1) and (1.1) , B1 B0 B2 B1 B0 1 = B + + x + + + x2 + 0 1!1! 0!2! 2!1! 1!2! 0!3! In order to holds this for arbitrary x , B1 B0 B2 B1 B0 B =1 , + =0 , + + =0 , 0 1!1! 0!2! 2!1! 1!2! 0!3! n The coefficient of x is as follows. Bn Bn-1 Bn-2 B1 B0 + + + + + = 0 n!1! ()n -1 !2! ()n-2 !3! 1!n ! 0!()n +1 ! Multiplying both sides by ()n +1 ! , Bn()n +1 ! Bn-1()n+1 ! Bn-2()n +1 ! B1()n +1 ! B0 + + + + + = 0 n !1! ()n-1 !2! ()n -2 !3! 1!n ! 0! Using binomial coefficiets, n +1Ｃn Bn + n +1Ｃn-1 Bn-1 + n +1Ｃn-2 Bn-2 + + n +1Ｃ 1 B1 + n +1Ｃ 0 B0 = 0 Replacing n +1 with n , (1.3) nnＣ -1 Bn-1 + nnＣ -2 Bn-2 + nnＣ -3 Bn-3 + + nＣ1 B1 + nＣ0 B0 = 0 Substituting n =2, 3, 4, for this one by one, 1 21Ｃ B + 20Ｃ B = 0 B = - 1 0 2 2 1 32Ｃ B + 31Ｃ B + 30Ｃ B = 0 B = 2 1 0 2 6 Thus, we obtain Bernoulli numbers.
    [Show full text]
  • 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.
    [Show full text]
  • Bernoulli Numbers and Polynomials from a More General Point of View
    ENTE PER LE NUOVE TECNOLOGIE, L’ENERGIA E L’AMBIENTE Serie Innovazione I BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW GIUSEPPE. DATTOLI ENEA - Divisione Fisica Applicata Centro Ricerche Frascati, Roma CLEMENTE CESARANO ULM University, Department of Mathematics, ULM, Germany SILVERIA LORENZUTTA ENEA -P Divisione Fisica Applicata Centro Ricerche “Ezio Clementel”, Bologna RT/1NN/2ooo/11 , Questo rapporto e stato preparato da: Servizio Edizioni Scientifiche - ENEA, Centro Ricerche Frascati, C.P. 65-00044 Frascati, Roma, Italia I contenuti tecnico-scientifici dei rapporti tecnici deH’ENEA rispecchiano I’opinione degli autori e non necessariamente quells dell’Ente. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW Riassunto Si applica il metodo dells funzione generatrice per introdurre nuove forme di numeri e polinomi di Bernoulli che vengono utilizzati per sviluppare e per calcolare somme parziali che coinvolgono polinomi a piti indici ed a piii variabili. Si sviluppano considerazioni analoghe per i polinomi ed i numeri di Eulero. BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW Abstract We apply the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further ciasses of partial sums involving generalized many index many variable polynomials. Analogous considerations are develo- ped for the Euler numbers and polynomials. Key words: numeri di Bernoulli, polinomi di Bernoulli, polinomi di Hermite, polinomi di AppEl, somme parziali INDICE 1- ~TRODUCTION .....................................................................................................p. 7 2- FINITE SUMS AND NEW CLASSES OF BERNOULLI NUMBERS .....................
    [Show full text]
  • Some Identities on the Poly-Genocchi Polynomials and Numbers
    Article Some Identities on the Poly-Genocchi Polynomials and Numbers Dmitry V. Dolgy 1 and Lee-Chae Jang 2,* 1 Kwangwoon Glocal Education Center, Kwangwoon University, Seoul 139-701, Korea; [email protected] 2 Graduate School of Education, Konkuk University, Seoul 143-701, Korea * Correspondence: [email protected] Received: 28 May 2020; Accepted: 9 June 2020; Published: 14 June 2020 Abstract: Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive various properties of them. Furthermore, we define unipoly Genocchi polynomials attached to an arithmetic function and investigate some identities of them. Keywords: polylogarithm functions; poly-Genocchi polynomials; unipoly functions; unipoly Genocchi polynomials MSC: 11B83; 11S80 1. Introduction The study of the generalized versions of Bernoulli and Euler polynomials and numbers was carried out in [1,2]. In recent years, various special polynomials and numbers regained the interest of mathematicians and quite a few results have been discovered. They include the Stirling numbers of the first and the second kind, central factorial numbers of the second kind, Bernoulli numbers of the second kind, Bernstein polynomials, Bell numbers and polynomials, central Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, Cauchy numbers, and others (see [3–8] and the references therein). We mention that the study of a generalized version of the special polynomials and numbers can be done also for the transcendental functions like hypergeometric ones.
    [Show full text]
  • Some Properties of Bernoulli Polynomials and Their Generalizations
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 746–751 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Some properties of Bernoulli polynomials and their generalizations Da-Qian Lu East China Normal University, Department of Mathematics, Shanghai 200241, PR China article info a b s t r a c t Article history: In this work, we investigate some well-known and new properties of the Bernoulli Received 18 July 2010 polynomials and their generalizations by using quasi-monomial, lowering operator and Accepted 21 December 2010 operational methods. Some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the (generalized) Keywords: Bernoulli polynomials. Quasi-monomial ' 2010 Elsevier Ltd. All rights reserved. Generating functions Recursion formulas Connection problems Bernoulli polynomials and numbers Apostol–Bernoulli polynomials and numbers 1. Introduction O O A polynomial pn.x/.n 2 N; x 2 C/ is said to be a quasi-monomial [1] whenever two operators M; P, called the multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that O Ppn.x/ D npn−1.x/; (1.1) O Mpn.x/ D pnC1.x/; (1.2) which can be combined to get the identity O O MPpn.x/ D npn.x/: (1.3) The classical Bernoulli polynomials Bn.x/ are defined by [2] 1 zexz X zn D Bn.x/ .jzj < 2π/; (1.4) ez − 1 nW nD0 and, consequently, the classical Bernoulli numbers Bn VD Bn.0/ can be obtained by using the generating function 1 z X zn D Bn : (1.5) ez − 1 nW nD0 Moreover, we have [3] n 1 1 Bn.0/ D .−1/ Bn.1/ D Bn .n 2 N0/: (1.6) 21−n − 1 2 E-mail address: [email protected].
    [Show full text]
  • On the Apostol-Bernoulli Polynomials
    CEJM 2(4) 2004 509{515 On the Apostol-Bernoulli Polynomials Qiu-Ming Luo¤ Department of Mathematics, Jiaozuo University, Jiaozuo City, Henan 454003, The People's Republic of China Received 19 July 2001; accepted 23 May 2003 Abstract: In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Paci¯c J. Math., 1 (1951), 161{167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications. °c Central European Science Journals. All rights reserved. Keywords: Bernoulli numbers, Bernoulli polynomials, Apostol-Bernoulli numbers, Apostol- Bernoulli polynomials, Gaussian hypergeometric functions, Stirling numbers of the second kind, Hurwitz Zeta functions, Lerch functional equation MSC (2000): Primary: 11B68; Secondary: 33C05, 11M35, 30E20 1 Introduction An analogue of the classical Bernoulli polynomials were de¯ned by T. M. Apostol (see [1]) when he studied the Lipschitz-Lerch Zeta functions. We call this polynomials the Apostol-Bernoulli polynomials. First we rewrite Apostol's de¯nitions below De¯nition 1.1. Apostol-Bernoulli polynomials Bn(x; ¸) are de¯ned by means of the generating function (see [1, p.165 (3.1)] or [4, p.83]) 1 zexz zn = B (x; ¸) ; (jz + ln ¸j < 2¼) (1) ¸ez ¡ 1 n n! n=0 X setting ¸ = 1 in (1), Bn(x) = Bn(x; 1) are classical Bernoulli polynomials. ¤ [email protected] 510 Q.-M. Luo / Central European Journal of Mathematics 2(4) 2004 509{515 De¯nition 1.2. Apostol-Bernoulli numbers Bn(¸) := Bn(0; ¸) are de¯ned by means of the generating function 1 z zn = B (¸) ; (jz + ln ¸j < 2¼) (2) ¸ez ¡ 1 n n! n=0 X setting ¸ = 1 in (2), Bn = Bn(1) are classical Bernoulli nunbers.
    [Show full text]
  • New Integral Representations of the Polylogarithm Function
    New integral representations of the polylogarithm function Djurdje Cvijović1 1 Atomic Physics Laboratory, Vinča Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Serbia. Abstract. Maximon has recently given an excellent summary of the properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Lis () z . The polylogarithm function appears in several fields of mathematics and in many physical problems. We, by making use of elementary arguments, deduce several new integral representations of the polylogarithm Lis () z for any complex z for which z < 1. Two are valid for all complex s, whenever Res > 1. The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is an positive integer. Our earlier established results on the integral representations for the Riemann zeta function ζ (2n + 1) , nN∈ , follow directly as corollaries of these representations. Keywords: Polylogarithms, integral representation, Riemann’s zeta function, Bernoulli polynomials 2000 Mathematics Subject Classification. Primary 11M99; Secondary 33E20. 1 1. Introduction Recently, Maximon [1] has given an excellent summary of the defining equations and properties of the Euler dilogarithm function and the frequently used generalizations of the dilogarithm, the most important among them being the polylogarithm function Lis () z . These include integral representations, series expansions, linear and quadratic transformations, functional relations and numerical values for special arguments. Motivated by this paper we have begun a systematic study of new integral representations for Lis () z since it appears that only half a dozen of them can be found in the literature.
    [Show full text]
  • Bernoulli Numbers and Polynomials
    Bernoulli Numbers and Polynomials T. Muthukumar [email protected] 17 Jun 2014 The sum of first n natural numbers 1; 2; 3; : : : ; n is n X n(n + 1) n2 n S (n) := m = = + : 1 2 2 2 m=1 This formula can be derived by noting that S1(n) = 1 + 2 + ::: + n S1(n) = n + (n − 1) + ::: + 1: Therefore, summing term-by-term, 2S1(n) = (n + 1) + ::: + (n + 1) = n(n + 1): | {z } n−times An alternate way of obtaining the above sum is by using the following two identity: (i)( m + 1)2 − m2 = 2m + 1 and, hence, n n X X 2 m = (m + 1)2 − m2 − n: m=1 m=1 (ii) n X (m + 1)2 − m2 = [22 − 12] + [32 − 22] + ::: + m=1 +[n2 − (n − 1)2] + [(n + 1)2 − n2] = (n + 1)2 − 1: 1 Thus, (n + 1)2 − (n + 1) n(n + 1) S (n) = = : 1 2 2 More generally, the sum of k-th power of first n natural numbers is de- noted as k k k Sk(n) := 1 + 2 + ::: + n : 0 Since a = 1, for any a, we have S0(n) = n. For k 2 N, one can compute Sk(n) using the identities: (i) k X k + 1 (m + 1)k+1 − mk+1 = mi i i=0 and, hence, n n k−1 n X X X X k + 1 (k + 1) mk = (m + 1)k+1 − mk+1 − mi: i m=1 m=1 i=0 m=1 Pn k+1 k+1 k+1 (ii) m=1 (m + 1) − m = (n + 1) − 1.
    [Show full text]