Various Old and New Results in Classical Arithmetic by Special Functions
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Liouville Numbers
Math 154 Notes 2 In these notes I’ll construct some explicit transcendental numbers. These numbers are examples of Liouville numbers. Let {ak} be a sequence of positive integers, and suppose that ak divides ak+1 for all k. We say that {ak} has moderate growth if there is some con- K stant K such that an+1 <K(an) for all n. Otherwise we say that {an} has immoderate growth. ∞ 1 Theorem 0.1 k P If {a } has immoderate growth then the number x = n=1 an is transcendental. Proof: We will suppose that P (x) = 0 for some polynomial P . There is some constant C1 such that P has degree at most C1 and all the coefficients of P are less than C1 in absolute value. Let xn = 1/a1 + ... +1/an. The divisibility condition guarantees that xn = hn/an, where hn is some integer. As long as P (xn) is nonzero, we have −C1 |P (xn) − P (x)| = |P (xn)|≥ (an) . (1) This just comes from the fact that P (xn) is a rational number whose denom- C1 inator is at most (an) . ′ On the other hand, there is some constant C2 such that |P (u)| < C2 for all u in the interval [x − 1,x + 1]. For n large, xn lies in this interval. Therefore, by the usual estimate that comes from integration, −1 |P (xn) − P (x)| <C2|xn − x| < 2C2an+1. (2) The last inequality comes from the fact that {1/an} decays faster than the sequence {1/2n} once n is large. -
Irrationality and Transcendence of Alternating Series Via Continued
Mathematical Assoc. of America American Mathematical Monthly 0:0 October 1, 2020 12:43 a.m. alternatingseries.tex page 1 Irrationality and Transcendence of Alternating Series Via Continued Fractions Jonathan Sondow Abstract. Euler gave recipes for converting alternating series of two types, I and II, into equiv- alent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that e, sin 1, and the pri- morial constant are irrational. Our main result is that, if a series of type II is equivalent to a sim- ple continued fraction, then the sum is transcendental and its irrationality measure exceeds 2. ℵℵ0 = We construct all 0 c such series and recover the transcendence of the Davison–Shallit and Cahen constants. Along the way, we mention π, the golden ratio, Fermat, Fibonacci, and Liouville numbers, Sylvester’s sequence, Pierce expansions, Mahler’s method, Engel series, and theorems of Lambert, Sierpi´nski, and Thue-Siegel-Roth. We also make three conjectures. 1. INTRODUCTIO. In a 1979 lecture on the Life and Work of Leonhard Euler, Andr´eWeil suggested “that our students of mathematics would profit much more from a study of Euler’s Introductio in Analysin Infinitorum, rather than of the available mod- ern textbooks” [17, p. xii]. The last chapter of the Introductio is “On Continued Frac- tions.” In it, after giving their form, Euler “next look[s] for an equivalent expression in the usual way of expressing fractions” and derives formulas for the convergents. He then converts a continued fraction into an equivalent alternating series, i.e., one whose partial sums equal the convergents. -
Fibonacci and -Lucas Numbers
Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2013, Article ID 107145, 7 pages http://dx.doi.org/10.1155/2013/107145 Research Article Incomplete -Fibonacci and -Lucas Numbers José L. Ramírez Instituto de Matematicas´ y sus Aplicaciones, Universidad Sergio Arboleda, Colombia Correspondence should be addressed to JoseL.Ram´ ´ırez; [email protected] Received 10 July 2013; Accepted 11 September 2013 Academic Editors: B. Li and W. van der Hoek Copyright © 2013 JoseL.Ram´ ´ırez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We define the incomplete k-Fibonacci and k-Lucas numbers; we study the recurrence relations and some properties of these numbers. 1. Introduction In [6],theauthorsdefinetheincompleteFibonacciandLucas -numbers. Also the authors define the incomplete bivariate Fibonacci numbers and their generalizations have many Fibonacci and Lucas -polynomials in [7]. interesting properties and applications to almost every field On the other hand, many kinds of generalizations of of science and art (e.g., see [1]). Fibonacci numbers are Fibonaccinumbershavebeenpresentedintheliterature.In defined by the recurrence relation particular, a generalization is the -Fibonacci Numbers. For any positive real number ,the-Fibonacci sequence, 0 =0, 1 =1, +1 = +−1,⩾1. (1) say {,}∈N, is defined recurrently by There exist a lot of properties about Fibonacci numbers. =0, =1, = + ,⩾1. In particular, there is a beautiful combinatorial identity to ,0 ,1 ,+1 , ,−1 (4) Fibonacci numbers [1] In [8], -Fibonacci numbers were found by studying ⌊(−1)/2⌋ −−1 = ∑ ( ). the recursive application of two geometrical transformations (2) =0 used in the four-triangle longest-edge (4TLE) partition. -
On H(X)-Fibonacci Octonion Polynomials, Ahmet Ipek & Kamil
Alabama Journal of Mathematics ISSN 2373-0404 39 (2015) On h(x)-Fibonacci octonion polynomials Ahmet Ipek˙ Kamil Arı Karamanoglu˘ Mehmetbey University, Karamanoglu˘ Mehmetbey University, Science Faculty of Kamil Özdag,˘ Science Faculty of Kamil Özdag,˘ Department of Mathematics, Karaman, Turkey Department of Mathematics, Karaman, Turkey In this paper, we introduce h(x)-Fibonacci octonion polynomials that generalize both Cata- lan’s Fibonacci octonion polynomials and Byrd’s Fibonacci octonion polynomials and also k- Fibonacci octonion numbers that generalize Fibonacci octonion numbers. Also we derive the Binet formula and and generating function of h(x)-Fibonacci octonion polynomial sequence. Introduction and Lucas numbers (Tasci and Kilic (2004)). Kiliç et al. (2006) gave the Binet formula for the generalized Pell se- To investigate the normed division algebras is greatly a quence. important topic today. It is well known that the octonions The investigation of special number sequences over H and O are the nonassociative, noncommutative, normed division O which are not analogs of ones over R and C has attracted algebra over the real numbers R. Due to the nonassociative some recent attention (see, e.g.,Akyigit,˘ Kösal, and Tosun and noncommutativity, one cannot directly extend various re- (2013), Halici (2012), Halici (2013), Iyer (1969), A. F. Ho- sults on real, complex and quaternion numbers to octonions. radam (1963) and Keçilioglu˘ and Akkus (2015)). While ma- The book by Conway and Smith (n.d.) gives a great deal of jority of papers in the area are devoted to some Fibonacci- useful background on octonions, much of it based on the pa- type special number sequences over R and C, only few of per of Coxeter et al. -
On Transcendental Numbers: New Results and a Little History
Axioms 2013, xx, 1-x; doi:10.3390/—— OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication On transcendental numbers: new results and a little history Solomon Marcus1 and Florin F. Nichita1,* 1 Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel: (+40) (0) 21 319 65 06; Fax: (+40) (0) 21 319 65 05 Received: xx / Accepted: xx / Published: xx Abstract: Attempting to create a general framework for studying new results on transcendental numbers, this paper begins with a survey on transcendental numbers and transcendence, it then presents several properties of the transcendental numbers e and π, and then it gives the proofs of new inequalities and identities for transcendental numbers. Also, in relationship with these topics, we study some implications for the theory of the Yang-Baxter equations, and we propose some open problems. Keywords: Euler’s relation, transcendental numbers; transcendental operations / functions; transcendence; Yang-Baxter equation Classification: MSC 01A05; 11D09; 11T23; 33B10; 16T25 arXiv:1502.05637v2 [math.HO] 5 Dec 2017 1. Introduction One of the most famous formulas in mathematics, the Euler’s relation: eπi +1=0 , (1) contains the transcendental numbers e and π, the imaginary number i, the constants 0 and 1, and (transcendental) operations. Beautiful, powerful and surprising, it has changed the mathematics forever. We will unveil profound aspects related to it, and we will propose a counterpart: ei π < e , (2) | − | Axioms 2013, xx 2 Figure 1. An interpretation of the inequality ei π < e . -
Transcendental Numbers
INTRODUCTION TO TRANSCENDENTAL NUMBERS VO THANH HUAN Abstract. The study of transcendental numbers has developed into an enriching theory and constitutes an important part of mathematics. This report aims to give a quick overview about the theory of transcen- dental numbers and some of its recent developments. The main focus is on the proof that e is transcendental. The Hilbert's seventh problem will also be introduced. 1. Introduction Transcendental number theory is a branch of number theory that concerns about the transcendence and algebraicity of numbers. Dated back to the time of Euler or even earlier, it has developed into an enriching theory with many applications in mathematics, especially in the area of Diophantine equations. Whether there is any transcendental number is not an easy question to answer. The discovery of the first transcendental number by Liouville in 1851 sparked up an interest in the field and began a new era in the theory of transcendental number. In 1873, Charles Hermite succeeded in proving that e is transcendental. And within a decade, Lindemann established the tran- scendence of π in 1882, which led to the impossibility of the ancient Greek problem of squaring the circle. The theory has progressed significantly in recent years, with answer to the Hilbert's seventh problem and the discov- ery of a nontrivial lower bound for linear forms of logarithms of algebraic numbers. Although in 1874, the work of Georg Cantor demonstrated the ubiquity of transcendental numbers (which is quite surprising), finding one or proving existing numbers are transcendental may be extremely hard. In this report, we will focus on the proof that e is transcendental. -
Alyssa Farrell
Recursive Patterns and Polynomials Alyssa Farrell Iowa State University MSM Creative Component Summer 2006 Heather Thompson, Major Professor Irvin Hentzel, Major Professor Patricia Leigh, Committee Member Farrell 2 Patterns have been studied for centuries. Some of these patterns have been graced with names of the people who are most remembered for their work with them or related topics. Among these are the Fibonacci numbers, Lucas numbers, Pascal’s Triangle, Chebyshev’s Polynomials, and Cardan Polynomials. Patterns are used as building blocks in education from newborn babies and young children all the way through high school and beyond into post-secondary work. We were introduced as early as infancy to learning to recognize patterns, meaning both similarities and differences, when learning colors, shapes, the alphabet, and numbers. As we grew, we were introduced to new mathematical patterns like learning to count by 2’s, or 5’s. As we were more formally introduced to shapes, we learned to recognize that a square has four sides the same length, but unlike a kite, the square has four angles that are 90 degrees. Then we learned to add, subtract, multiply, and divide. All of these necessary operations were taught using a pattern-approach. Do you remember learning to read? We were taught to recognize patterns of letters which made certain sounds and that when certain sounds were strung together they made words. These patterns continued to be used as we developed higher in our math education. Teachers in junior high or middle school introduced us to an abstract idea of letting letters represent numbers as variables. -
Solving Recurrence Equations
Great Theoretical Ideas In Computer Science V. Adamchik CS 15-251 Spring 2010 D. Sleator Solve in integers Lecture 8 Feb 04, 2010 Carnegie Mellon University Recurrences and Continued Fractions x1 + x2 +…+ x5 = 40 xkrk; yk=xk – k y1 + y2 +…+ y5 = 25 y ≥ 0; k 29 4 Partitions Solve in integers Find the number of ways to partition the integer n x + y + z = 11 3 = 1+1+1=1+2 xr0; yb3; zr0 4=1+1+1+1=1+1+2=1+3=2+2 2 3 x1 2x2 3x3 ... nxn n [X11] 1 x x x (1 -x) 2 1 (1 -x)(1 -x2)(1 -x3)...(1 -xn) Leonardo Fibonacci The rabbit reproduction model •A rabbit lives forever In 1202, Fibonacci has become •The population starts as a single newborn interested in rabbits and what they do pair •Every month, each productive pair begets a new pair which will become productive after 2 months old Fn= # of rabbit pairs at the beginning of the nth month month 1 2 3 4 5 6 7 rabbits 1 1 2 3 5 8 13 1 F =0, F =1, 0 1 WARNING!!!! Fn=Fn-1+Fn-2 for n≥2 This lecture has What is a closed form explicit mathematical formula for Fn? content that can be shocking to some students. n n-1 n-2 Characteristic Equation - - = 0 n-2 2 Characteristic Fn=Fn-1+Fn-2 iff ( - - 1) = 0 equation Consider solutions of the form: iff 2 - - 1 = 0 n 1 5 Fn= for some (unknown) constant ≠ 0 2 = , or = -1/ must satisfy: (“phi”) is the golden ratio n - n-1 - n-2 = 0 a,b a n + b (-1/ )n = , or = -(1/ ) satisfies the inductive condition So for all these values of the Adjust a and b to fit the base inductive condition is satisfied: conditions. -
Certain Combinatorial Results on Two Variable Hybrid Fibonacci Polynomials
International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 2 (2017), pp. 603–613 © Research India Publications http://www.ripublication.com/ijcam.htm Certain Combinatorial Results on Two variable Hybrid Fibonacci Polynomials R. Rangarajan, Honnegowda C.K.1, Shashikala P. Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru - 570 006, INDIA. Abstract Recently many researchers in combinatorics have contributed to the development of one variable polynomials related to hypergeometric family. Here a class of two variable hybrid Fibonacci polynomials is defined and proved certain combinatorial results. AMS subject classification: 05A19, 11B39 and 33C45. Keywords: Combinatorial Identities, Fibonacci Numbers and Hyper Geometric Functions in one or more Variables. 1. Introduction The natural and the most beautiful three term recurrence relation is Fn+1 = Fn + Fn−1, F1 = 1,F2 = 1, where Fn is the nth Fibonacci number [1, 2, 3, 13]. There are many one variable generalizations of Fn related to hypergeometric family [3, 4]. Two important such generalizations are Catalan polynomials and Jacobsthal polynomials denoted by (C) (J ) fn (x) and fn (x) which exhibits many interesting combinatorial properties [4, 6, 7, 8, 9, 10, 12]. 1Coressoponding author. 604 R. Rangarajan, Honnegowda C.K., and Shashikala P. Two hypergeometric polynomials, namely, Un(x) and Vn(x) called Tchebyshev poly- nomials of second and third kind are given by n+1 n+1 1 2 2 Un(x) = √ x + x − 1 − x − x − 1 2 x2 − 1 3 1 − x = (n + 1) F −n, n + 2; ; 2 1 2 2 and = − Vn(x) Un+1(x) Un(x) 1 1 − x = F −n, n + 1; ; 2 1 2 2 which have many applications in both pure and applied mathematics [5, 6, 7, 8, 9, 10, 3 3 11, 12]. -
A Computable Absolutely Normal Liouville Number
MATHEMATICS OF COMPUTATION Volume 84, Number 296, November 2015, Pages 2939–2952 http://dx.doi.org/10.1090/mcom/2964 Article electronically published on April 24, 2015 A COMPUTABLE ABSOLUTELY NORMAL LIOUVILLE NUMBER VERONICA´ BECHER, PABLO ARIEL HEIBER, AND THEODORE A. SLAMAN Abstract. We give an algorithm that computes an absolutely normal Liou- ville number. 1. The main result The set of Liouville numbers is {x ∈ R \ Q : ∀k ∈ N, ∃q ∈ N,q>1and||qx|| <q−k} where ||x|| =min{|x−m| : m ∈ Z} is the distance of a real number x to the nearest −k! integer and other notation is as usual. Liouville’s constant, k≥1 10 ,isthe standard example of a Liouville number. Though uncountable, the set of Liouville numbers is small, in fact, it is null, both in Lebesgue measure and in Hausdorff dimension (see [6]). We say that a base is an integer s greater than or equal to 2. A real number x is normal to base s if the sequence (sjx : j ≥ 0) is uniformly distributed in the unit interval modulo one. By Weyl’s Criterion [11], x is normal to base s if and only if certain harmonic sums associated with (sjx : j ≥ 0) grow slowly. Absolute normality is normality to every base. Bugeaud [6] established the existence of absolutely normal Liouville numbers by means of an almost-all argument for an appropriate measure due to Bluhm [3, 4]. The support of this measure is a perfect set, which we call Bluhm’s fractal, all of whose irrational elements are Liouville numbers. -
Generalized Fibonacci-Type Polynomials
GENERALIZED FIBONACCI-TYPE POLYNOMIALS Yashwant K. Panwar 1 , G. P. S. Rathore 2 , Richa Chawla 3 1 Department of Mathematics, Mandsaur University, Mandsaur, (India) 2 Department of Mathematical Sciences, College of Horticulture, Mandsaur,(India) 3 School of Studies in Mathematics, Vikram University, Ujjain, (India) ABSTRACT In this study, we present Fibonacci-type polynomials. We have used their Binet’s formula to derive the identities. The proofs of the main theorems are based on simple algebra and give several interesting properties involving them. Mathematics Subject Classification: 11B39, 11B37 Keywords: Generalized Fibonacci polynomials, Binet’s formula, Generating function. I. INTRODUCTION Fibonacci polynomials are a great importance in mathematics. Large classes of polynomials can be defined by Fibonacci-like recurrence relation and yield Fibonacci numbers [15]. Such polynomials, called the Fibonacci polynomials, were studied in 1883 by the Belgian Mathematician Eugene Charles Catalan and the German Mathematician E. Jacobsthal. The polynomials fxn ()studied by Catalan are defined by the recurrence relation fn21()()() x xf n x f n x (1.1) where f12( x ) 1, f ( x ) x , and n 3 . Notice that fFnn(1) , the nth Fibonacci number. The Fibonacci polynomials studied by Jacobsthal were defined by Jn()()() x J n12 x xJ n x (1.2) where J12( x ) 1 J ( x ) , and . The Pell polynomials pxn ()are defined by pn( x ) 2 xp n12 ( x ) p n ( x ) (1.3) where p01( x ) 0, p ( x ) 1, and n 2 . The Lucas polynomials lxn (), originally studied in 1970 by Bicknell, are defined by ln()()() x xl n12 x l n x (1.4) where l01( x ) 2, l ( x ) x , and n 2 . -
Liouville Numbers
Liouville Numbers Manuel Eberl April 17, 2016 Abstract In this work, we define the concept of Liouville numbers as well as the standard construction to obtain Liouville numbers and we prove their most important properties: irrationality and transcendence. This is historically interesting since Liouville numbers constructed in the standard way where the first numbers that were proven to be transcendental. The proof is very elementary and requires only stan- dard arithmetic and the Mean Value Theorem for polynomials and the boundedness of polynomials on compact intervals. Contents 1 Liouville Numbers1 1.1 Preliminary lemmas.......................1 1.2 Definition of Liouville numbers.................3 1.3 Standard construction for Liouville numbers..........7 1 Liouville Numbers 1.1 Preliminary lemmas theory Liouville-Numbers-Misc imports Complex-Main ∼∼=src=HOL=Library=Polynomial begin We will require these inequalities on factorials to show properties of the standard construction later. lemma fact-ineq: n ≥ 1 =) fact n + k ≤ fact (n + k) proof (induction k) case (Suc k) from Suc have fact n + Suc k ≤ fact (n + k) + 1 by simp also from Suc have ::: ≤ fact (n + Suc k) by simp finally show ?case . 1 qed simp-all lemma Ints-setsum: assumes Vx: x 2 A =) f x 2 ZZ shows setsum f A 2 ZZ by (cases finite A; insert assms; induction A rule: finite-induct) (auto intro!: Ints-add) lemma suminf-split-initial-segment 0: summable (f :: nat ) 0a::real-normed-vector) =) suminf f = (P n: f (n + k + 1 )) + setsum f f::kg by (subst suminf-split-initial-segment[of