L-Series and Hurwitz Zeta Functions Associated with the Universal Formal Group
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The Lerch Zeta Function and Related Functions
The Lerch Zeta Function and Related Functions Je↵ Lagarias, University of Michigan Ann Arbor, MI, USA (September 20, 2013) Conference on Stark’s Conjecture and Related Topics , (UCSD, Sept. 20-22, 2013) (UCSD Number Theory Group, organizers) 1 Credits (Joint project with W. C. Winnie Li) J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function I. Zeta Integrals, Forum Math, 24 (2012), 1–48. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function II. Analytic Continuation, Forum Math, 24 (2012), 49–84. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function III. Polylogarithms and Special Values, preprint. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function IV. Two-variable Hecke operators, in preparation. Work of J. C. Lagarias is partially supported by NSF grants DMS-0801029 and DMS-1101373. 2 Topics Covered Part I. History: Lerch Zeta and Lerch Transcendent • Part II. Basic Properties • Part III. Multi-valued Analytic Continuation • Part IV. Consequences • Part V. Lerch Transcendent • Part VI. Two variable Hecke operators • 3 Part I. Lerch Zeta Function: History The Lerch zeta function is: • e2⇡ina ⇣(s, a, c):= 1 (n + c)s nX=0 The Lerch transcendent is: • zn Φ(s, z, c)= 1 (n + c)s nX=0 Thus ⇣(s, a, c)=Φ(s, e2⇡ia,c). 4 Special Cases-1 Hurwitz zeta function (1882) • 1 ⇣(s, 0,c)=⇣(s, c):= 1 . (n + c)s nX=0 Periodic zeta function (Apostol (1951)) • e2⇡ina e2⇡ia⇣(s, a, 1) = F (a, s):= 1 . ns nX=1 5 Special Cases-2 Fractional Polylogarithm • n 1 z z Φ(s, z, 1) = Lis(z)= ns nX=1 Riemann zeta function • 1 ⇣(s, 0, 1) = ⇣(s)= 1 ns nX=1 6 History-1 Lipschitz (1857) studies general Euler integrals including • the Lerch zeta function Hurwitz (1882) studied Hurwitz zeta function. -
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. -
[Math.CA] 1 Jul 1992 Napiain.Freape Ecnlet Can We Example, for Applications
Convolution Polynomials Donald E. Knuth Computer Science Department Stanford, California 94305–2140 Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F (x), F (x), F (x),... forms a convolution family if F (x) has degree n 0 1 2 n ≤ and if the convolution condition F (x + y)= F (x)F (y)+ F − (x)F (y)+ + F (x)F − (y)+ F (x)F (y) n n 0 n 1 1 · · · 1 n 1 0 n holds for all x and y and for all n 0. Many such families are known, and they appear frequently ≥ n in applications. For example, we can let Fn(x)= x /n!; the condition (x + y)n n xk yn−k = n! k! (n k)! kX=0 − is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial x coefficient n ; the corresponding identity n x + y x y = n k n k Xk=0 − is commonly called Vandermonde’s convolution. How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 n 4: ≤ ≤ F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! arXiv:math/9207221v1 [math.CA] 1 Jul 1992 conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}] ==Series[Sum[F[k,x]F[n-k,y],{k,0,n}],{x,0,n},{y,0,n}]] Solve[Table[conv[n],{n,0,4}], [Flatten[Table[f[i,j],{i,0,4},{j,0,4}]]]] Mathematica replies that the F ’s are either identically zero or the coefficients of Fn(x) = fn0 + f x + f x2 + + f xn /n! satisfy n1 n2 · · · nn f00 = 1 , f10 = f20 = f30 = f40 = 0 , 2 3 f22 = f11 , f32 = 3f11f21 , f33 = f11 , 2 2 4 f42 = 4f11f31 + 3f21 , f43 = 6f11f21 , f44 = f11 . -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
1 Evaluation of Series with Hurwitz and Lerch Zeta Function Coefficients by Using Hankel Contour Integrals. Khristo N. Boyadzhi
Evaluation of series with Hurwitz and Lerch zeta function coefficients by using Hankel contour integrals. Khristo N. Boyadzhiev Abstract. We introduce a new technique for evaluation of series with zeta coefficients and also for evaluation of certain integrals involving the logGamma function. This technique is based on Hankel integral representations of the Hurwitz zeta, the Lerch Transcendent, the Digamma and logGamma functions. Key words: Hankel contour, Hurwitz zeta function, Lerch Transcendent, Euler constant, Digamma function, logGamma integral, Barnes function. 2000 Mathematics Subject Classification: Primary 11M35; Secondary 33B15, 40C15. 1. Introduction. The Hurwitz zeta function is defined for all by , (1.1) and has the integral representation: . (1.2) When , it turns into Riemann’s zeta function, . In this note we present a new method for evaluating the series (1.3) and (1.4) 1 in a closed form. The two series have received a considerable attention since Srivastava [17], [18] initiated their systematic study in 1988. Many interesting results were obtained consequently by Srivastava and Choi (for instance, [6]) and were collected in their recent book [19]. Fundamental contributions to this theory and independent evaluations belong also to Adamchik [1] and Kanemitsu et al [13], [15], [16], Hashimoto et al [12]. For some recent developments see [14]. The technique presented here is very straightforward and applies also to series with the Lerch Transcendent [8]: , (1.5) in the coefficients. For example, we evaluate here in a closed form the series (1.6) The evaluation of (1.3) and (1.4) requires zeta values for positive and negative integers . We use a representation of in terms of a Hankel integral, which makes it possible to represent the values for positive and negative integers by the same type of integral. -
The Riemann and Hurwitz Zeta Functions, Apery's Constant and New
The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving ζ(2k) Cezar Lupu1 1Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra, Combinatorics and Geometry Graduate Student Research Seminar, February 2, 2017, Pittsburgh, PA A quick overview of the Riemann zeta function. The Riemann zeta function is defined by 1 X 1 ζ(s) = ; Re s > 1: ns n=1 Originally, Riemann zeta function was defined for real arguments. Also, Euler found another formula which relates the Riemann zeta function with prime numbrs, namely Y 1 ζ(s) = ; 1 p 1 − ps where p runs through all primes p = 2; 3; 5;:::. A quick overview of the Riemann zeta function. Moreover, Riemann proved that the following ζ(s) satisfies the following integral representation formula: 1 Z 1 us−1 ζ(s) = u du; Re s > 1; Γ(s) 0 e − 1 Z 1 where Γ(s) = ts−1e−t dt, Re s > 0 is the Euler gamma 0 function. Also, another important fact is that one can extend ζ(s) from Re s > 1 to Re s > 0. By an easy computation one has 1 X 1 (1 − 21−s )ζ(s) = (−1)n−1 ; ns n=1 and therefore we have A quick overview of the Riemann function. 1 1 X 1 ζ(s) = (−1)n−1 ; Re s > 0; s 6= 1: 1 − 21−s ns n=1 It is well-known that ζ is analytic and it has an analytic continuation at s = 1. At s = 1 it has a simple pole with residue 1. -
Multiple Hurwitz Zeta Functions
Proceedings of Symposia in Pure Mathematics Multiple Hurwitz Zeta Functions M. Ram Murty and Kaneenika Sinha Abstract. After giving a brief overview of the theory of multiple zeta func- tions, we derive the analytic continuation of the multiple Hurwitz zeta function X 1 ζ(s , ..., s ; x , ..., x ):= 1 r 1 r s s (n1 + x1) 1 ···(nr + xr) r n1>n2>···>nr ≥1 using the binomial theorem and Hartogs’ theorem. We also consider the cog- nate multiple L-functions, X χ (n )χ (n ) ···χ (n ) L(s , ..., s ; χ , ..., χ )= 1 1 2 2 r r , 1 r 1 r s1 s2 sr n n ···nr n1>n2>···>nr≥1 1 2 where χ1, ..., χr are Dirichlet characters of the same modulus. 1. Introduction In a fundamental paper written in 1859, Riemann [34] introduced his celebrated zeta function that now bears his name and indicated how it can be used to study the distribution of prime numbers. This function is defined by the Dirichlet series ∞ 1 ζ(s)= ns n=1 in the half-plane Re(s) > 1. Riemann proved that ζ(s) extends analytically for all s ∈ C, apart from s = 1 where it has a simple pole with residue 1. He also established the remarkable functional equation − − s s −(1−s)/2 1 s π 2 ζ(s)Γ = π ζ(1 − s)Γ 2 2 and made the famous conjecture (now called the Riemann hypothesis) that if ζ(s)= 1 0and0< Re(s) < 1, then Re(s)= 2 . This is still unproved. In 1882, Hurwitz [20] defined the “shifted” zeta function, ζ(s; x)bytheseries ∞ 1 (n + x)s n=0 for any x satisfying 0 <x≤ 1. -
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. -
Skew and Infinitary Formal Power Series
Appeared in: ICALP’03, c Springer Lecture Notes in Computer Science, vol. 2719, pages 426-438. Skew and infinitary formal power series Manfred Droste and Dietrich Kuske⋆ Institut f¨ur Algebra, Technische Universit¨at Dresden, D-01062 Dresden, Germany {droste,kuske}@math.tu-dresden.de Abstract. We investigate finite-state systems with costs. Departing from classi- cal theory, in this paper the cost of an action does not only depend on the state of the system, but also on the time when it is executed. We first characterize the terminating behaviors of such systems in terms of rational formal power series. This generalizes a classical result of Sch¨utzenberger. Using the previous results, we also deal with nonterminating behaviors and their costs. This includes an extension of the B¨uchi-acceptance condition from finite automata to weighted automata and provides a characterization of these nonter- minating behaviors in terms of ω-rational formal power series. This generalizes a classical theorem of B¨uchi. 1 Introduction In automata theory, Kleene’s fundamental theorem [17] on the coincidence of regular and rational languages has been extended in several directions. Sch¨utzenberger [26] showed that the formal power series (cost functions) associated with weighted finite automata over words and an arbitrary semiring for the weights, are precisely the rational formal power series. Weighted automata have recently received much interest due to their applications in image compression (Culik II and Kari [6], Hafner [14], Katritzke [16], Jiang, Litow and de Vel [15]) and in speech-to-text processing (Mohri [20], [21], Buchsbaum, Giancarlo and Westbrook [4]). -
Euler and His Work on Infinite Series
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously. -
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. -
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.