Mathematical Constants and Sequences
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Generalizations of Euler Numbers and Polynomials 1
GENERALIZATIONS OF EULER NUMBERS AND POLYNOMIALS QIU-MING LUO AND FENG QI Abstract. In this paper, the concepts of Euler numbers and Euler polyno- mials are generalized, and some basic properties are investigated. 1. Introduction It is well-known that the Euler numbers and polynomials can be defined by the following definitions. Definition 1.1 ([1]). The Euler numbers Ek are defined by the following expansion t ∞ 2e X Ek = tk, |t| ≤ π. (1.1) e2t + 1 k! k=0 In [4, p. 5], the Euler numbers is defined by t/2 ∞ n 2n 2e t X (−1) En t = sech = , |t| ≤ π. (1.2) et + 1 2 (2n)! 2 n=0 Definition 1.2 ([1, 4]). The Euler polynomials Ek(x) for x ∈ R are defined by xt ∞ 2e X Ek(x) = tk, |t| ≤ π. (1.3) et + 1 k! k=0 It can also be shown that the polynomials Ei(t), i ∈ N, are uniquely determined by the following two properties 0 Ei(t) = iEi−1(t),E0(t) = 1; (1.4) i Ei(t + 1) + Ei(t) = 2t . (1.5) 2000 Mathematics Subject Classification. 11B68. Key words and phrases. Euler numbers, Euler polynomials, generalization. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China. This paper was typeset using AMS-LATEX. 1 2 Q.-M. LUO AND F. QI Euler polynomials are related to the Bernoulli numbers. For information about Bernoulli numbers and polynomials, please refer to [1, 2, 3, 4]. -
POWER FIBONACCI SEQUENCES 1. Introduction Let G Be a Bi-Infinite Integer Sequence Satisfying the Recurrence Relation G If
POWER FIBONACCI SEQUENCES JOSHUA IDE AND MARC S. RENAULT Abstract. We examine integer sequences G satisfying the Fibonacci recurrence relation 2 3 Gn = Gn−1 + Gn−2 that also have the property that G ≡ 1; a; a ; a ;::: (mod m) for some modulus m. We determine those moduli m for which these power Fibonacci sequences exist and the number of such sequences for a given m. We also provide formulas for the periods of these sequences, based on the period of the Fibonacci sequence F modulo m. Finally, we establish certain sequence/subsequence relationships between power Fibonacci sequences. 1. Introduction Let G be a bi-infinite integer sequence satisfying the recurrence relation Gn = Gn−1 +Gn−2. If G ≡ 1; a; a2; a3;::: (mod m) for some modulus m, then we will call G a power Fibonacci sequence modulo m. Example 1.1. Modulo m = 11, there are two power Fibonacci sequences: 1; 8; 9; 6; 4; 10; 3; 2; 5; 7; 1; 8 ::: and 1; 4; 5; 9; 3; 1; 4;::: Curiously, the second is a subsequence of the first. For modulo 5 there is only one such sequence (1; 3; 4; 2; 1; 3; :::), for modulo 10 there are no such sequences, and for modulo 209 there are four of these sequences. In the next section, Theorem 2.1 will demonstrate that 209 = 11·19 is the smallest modulus with more than two power Fibonacci sequences. In this note we will determine those moduli for which power Fibonacci sequences exist, and how many power Fibonacci sequences there are for a given modulus. -
Generalized Modular Forms Representable As Eta Products
Modular Forms Representable As Eta Products A Dissertation Submitted to the Temple University Graduate Board in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY by Wissam Raji August, 2006 iii c by Wissam Raji August, 2006 All Rights Reserved iv ABSTRACT Modular Forms Representable As Eta Products Wissam Raji DOCTOR OF PHILOSOPHY Temple University, August, 2006 Professor Marvin Knopp, Chair In this dissertation, we discuss modular forms that are representable as eta products and generalized eta products . Eta products appear in many areas of mathematics in which algebra and analysis overlap. M. Newman [15, 16] published a pair of well-known papers aimed at using eta-product to construct forms on the group Γ0(n) with the trivial multiplier system. Our work here divides into three related areas. The first builds upon the work of Siegel [23] and Rademacher [19] to derive modular transformation laws for functions defined as eta products (and related products). The second continues work of Kohnen and Mason [9] that shows that, under suitable conditions, a generalized modular form is an eta product or generalized eta product and thus a classical modular form. The third part of the dissertation applies generalized eta-products to rederive some arithmetic identities of H. Farkas [5, 6]. v ACKNOWLEDGEMENTS I would like to thank all those who have helped in the completion of my thesis. To God, the beginning and the end, for all His inspiration and help in the most difficult days of my life, and for all the people listed below. To my advisor, Professor Marvin Knopp, a great teacher and inspirer whose support and guidance were crucial for the completion of my work. -
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. -
An Introduction to Mathematical Modelling
An Introduction to Mathematical Modelling Glenn Marion, Bioinformatics and Statistics Scotland Given 2008 by Daniel Lawson and Glenn Marion 2008 Contents 1 Introduction 1 1.1 Whatismathematicalmodelling?. .......... 1 1.2 Whatobjectivescanmodellingachieve? . ............ 1 1.3 Classificationsofmodels . ......... 1 1.4 Stagesofmodelling............................... ....... 2 2 Building models 4 2.1 Gettingstarted .................................. ...... 4 2.2 Systemsanalysis ................................. ...... 4 2.2.1 Makingassumptions ............................. .... 4 2.2.2 Flowdiagrams .................................. 6 2.3 Choosingmathematicalequations. ........... 7 2.3.1 Equationsfromtheliterature . ........ 7 2.3.2 Analogiesfromphysics. ...... 8 2.3.3 Dataexploration ............................... .... 8 2.4 Solvingequations................................ ....... 9 2.4.1 Analytically.................................. .... 9 2.4.2 Numerically................................... 10 3 Studying models 12 3.1 Dimensionlessform............................... ....... 12 3.2 Asymptoticbehaviour ............................. ....... 12 3.3 Sensitivityanalysis . ......... 14 3.4 Modellingmodeloutput . ....... 16 4 Testing models 18 4.1 Testingtheassumptions . ........ 18 4.2 Modelstructure.................................. ...... 18 i 4.3 Predictionofpreviouslyunuseddata . ............ 18 4.3.1 Reasonsforpredictionerrors . ........ 20 4.4 Estimatingmodelparameters . ......... 20 4.5 Comparingtwomodelsforthesamesystem . ......... -
![Example 10.8 Testing the Steady-State Approximation. ⊕ a B C C a B C P + → → + → D Dt K K K K K K [ ]](https://docslib.b-cdn.net/cover/6323/example-10-8-testing-the-steady-state-approximation-a-b-c-c-a-b-c-p-d-dt-k-k-k-k-k-k-156323.webp)
Example 10.8 Testing the Steady-State Approximation. ⊕ a B C C a B C P + → → + → D Dt K K K K K K [ ]
Example 10.8 Testing the steady-state approximation. Å The steady-state approximation contains an apparent contradiction: we set the time derivative of the concentration of some species (a reaction intermediate) equal to zero — implying that it is a constant — and then derive a formula showing how it changes with time. Actually, there is no contradiction since all that is required it that the rate of change of the "steady" species be small compared to the rate of reaction (as measured by the rate of disappearance of the reactant or appearance of the product). But exactly when (in a practical sense) is this approximation appropriate? It is often applied as a matter of convenience and justified ex post facto — that is, if the resulting rate law fits the data then the approximation is considered justified. But as this example demonstrates, such reasoning is dangerous and possible erroneous. We examine the mechanism A + B ¾¾1® C C ¾¾2® A + B (10.12) 3 C® P (Note that the second reaction is the reverse of the first, so we have a reversible second-order reaction followed by an irreversible first-order reaction.) The rate constants are k1 for the forward reaction of the first step, k2 for the reverse of the first step, and k3 for the second step. This mechanism is readily solved for with the steady-state approximation to give d[A] k1k3 = - ke[A][B] with ke = (10.13) dt k2 + k3 (TEXT Eq. (10.38)). With initial concentrations of A and B equal, hence [A] = [B] for all times, this equation integrates to 1 1 = + ket (10.14) [A] A0 where A0 is the initial concentration (equal to 1 in work to follow). -
Differentiation Rules (Differential Calculus)
Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. -
Grade 7/8 Math Circles the Scale of Numbers Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero. -
Maths Secrets of Simpsons Revealed in New Book
MONDAY 7 OCTOBER 2013 WWW.THEDAY.CO.UK Maths secrets of Simpsons revealed in new book The most successful TV show of all time is written by a team of brilliant ‘mathletes’, says writer Simon Singh, and full of obscure mathematical jokes. Can numbers really be all that funny? MATHEMATICS Nerd hero: The smartest girl in Springfield was created by a team of maths wizards. he world’s most popular cartoon a perfect number, a narcissistic number insist that their love of maths contrib- family has a secret: their lines are and a Mersenne Prime. utes directly to the more obvious humour written by a team of expert mathema- Another of these maths jokes – a black- that has made the show such a hit. Turn- Tticians – former ‘mathletes’ who are board showing 398712 + 436512 = 447212 ing intuitions about comedy into concrete as happy solving differential equa- – sent shivers down Simon Singh’s spine. jokes is like wrestling mathematical tions as crafting jokes. ‘I was so shocked,’ he writes, ‘I almost hunches into proofs and formulas. Comedy Now, science writer Simon Singh has snapped my slide rule.’ The numbers are and maths, says Cohen, are both explora- revealed The Simpsons’ secret math- a fake exception to a famous mathemati- tions into the unknown. ematical formula in a new book*. He cal rule known as Fermat’s Last Theorem. combed through hundreds of episodes One episode from 1990 features a Mathletes and trawled obscure internet forums to teacher making a maths joke to a class of Can maths really be funny? There are many discover that behind the show’s comic brilliant students in which Bart Simpson who will think comparing jokes to equa- exterior lies a hidden core of advanced has been accidentally included. -
Perfect Gaussian Integer Sequences of Arbitrary Length Soo-Chang Pei∗ and Kuo-Wei Chang† National Taiwan University∗ and Chunghwa Telecom†
Perfect Gaussian Integer Sequences of Arbitrary Length Soo-Chang Pei∗ and Kuo-Wei Chang† National Taiwan University∗ and Chunghwa Telecom† m Objectives N = p or N = p using Legendre N = pq where p and q are coprime Conclusion sequence and Gauss sum To construct perfect Gaussian integer sequences Simple zero padding method: We propose several methods to generate zero au- of arbitrary length N: Legendre symbol is defined as 1) Take a ZAC from p and q. tocorrelation sequences in Gaussian integer. If the 8 2 2) Interpolate q − 1 and p − 1 zeros to these signals sequence length is prime number, we can use Leg- • Perfect sequences are sequences with zero > 1, if ∃x, x ≡ n(mod N) n ! > autocorrelation. = < 0, n ≡ 0(mod N) to get two signals of length N. endre symbol and provide more degree of freedom N > 3) Convolution these two signals, then we get a than Yang’s method. If the sequence is composite, • Gaussian integer is a number in the form :> −1, otherwise. ZAC. we develop a general method to construct ZAC se- a + bi where a and b are integer. And the Gauss sum is defined as N−1 ! Using the idea of prime-factor algorithm: quences. Zero padding is one of the special cases of • A Perfect Gaussian sequence is a perfect X n −2πikn/N G(k) = e Recall that DFT of size N = N1N2 can be done by this method, and it is very easy to implement. sequence that each value in the sequence is a n=0 N taking DFT of size N1 and N2 seperately[14]. -
On the Normality of Numbers
ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. -
The Aliquot Constant We first Show the Following Result on the Geometric Mean for the Ordi- Nary Sum of Divisors Function
THE ALIQUOT CONSTANT WIEB BOSMA AND BEN KANE Abstract. The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant λ when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded. 1. Introduction This paper is concerned with the average growth of aliquot sequences. An aliquot sequence is a sequence a0, a1, a2,... of positive integers ob- tained by iteration of the sum-of-aliquot-divisors function s, which is defined for n> 1 by s(n)= d. d|n Xd<n The aliquot sequence with starting value a0 is then equal to 2 a0, a1 = s(a0), a2 = s(a1)= s (a0),... ; k we will say that the sequence terminates (at 1) if ak = s (a0)=1 for some k ≥ 0. The sequence cycles (or is said to end in a cycle) k l 0 if s (a0) = s (a0) for some k,l with 0 ≤ l<k, where s (n) = n by arXiv:0912.3660v1 [math.NT] 18 Dec 2009 definition. Note that s is related to the ordinary sum-of-divisors function σ, with σ(n)= d|n d, by s(n)= σ(n) − n for integers n> 1.