DOCSLIB.ORG

Counting Prime Polynomials and Measuring Complexity and Similarity of Information

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Counting Prime Polynomials and
Measuring Complexity and Similarity of
Information

by
Niko Rebenich
B.Eng., University of Victoria, 2007 M.A.Sc., University of Victoria, 2012

A Dissertation Submitted in Partial Fulꢀllment of the
Requirements for the Degree of

DOCTOR OF PHILOSOPHY in the Department of Electrical and Computer Engineering

© Niko Rebenich, 2016
University of Victoria

All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii

Counting Prime Polynomials and
Measuring Complexity and Similarity of
Information

by
Niko Rebenich
B.Eng., University of Victoria, 2007 M.A.Sc., University of Victoria, 2012

Supervisory Committee Dr. Stephen Neville, Co-supervisor (Department of Electrical and Computer Engineering)

Dr. T. Aaron Gulliver, Co-supervisor (Department of Electrical and Computer Engineering)

Dr. Venkatesh Srinivasan, Outside Member (Department of Computer Science) iii

Supervisory Committee

Dr. Stephen Neville, Co-supervisor (Department of Electrical and Computer Engineering)

Dr. T. Aaron Gulliver, Co-supervisor (Department of Electrical and Computer Engineering)

Dr. Venkatesh Srinivasan, Outside Member (Department of Computer Science)

ABSTRACT

This dissertation explores an analogue of the prime number theorem for polynomials over ꢀnite ꢀelds as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Speciꢀcally, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational eꢁort. In this context asymptotic series expansions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. The expansion formulas developed are general and have applications in numerous areas other than the enumeration of prime polynomials.
A bijection between the equivalence classes of aperiodic necklaces and monic prime polynomials is utilized to derive an asymptotic bound on the maximal T- complexity value of a string. Furthermore, the statistical behaviour of uniform random sequences that are factored via the T-transform are investigated, and an accurate probabilistic model for short necklace factors is presented.
Finally, a T-complexity based conditional string complexity measure is proposed and used to deꢀne the normalized T-complexity distance that measures similarity between strings. The T-complexity distance is proven to not be a metric. However, the measure can be computed in linear time and space making it a suitable choice for large data sets. iv

Contents

Supervisory Committee Abstract

ii iii iv

Table of Contents List of Tables

vii ix

List of Figures List of Nomenclature Acknowledgements Dedication

xi xv xvi

1 Introduction

1

34

1.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Algebraic and Number Theory Background

6

6799

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Finite Field Extensions . . . . . . . . . . . . . . . . . . . . . . .
2.4 Primitive Roots of Unity and Cyclotomic Cosets . . . . . . . . . . . . . 12 2.5 Monic Irreducible Polynomials and Necklaces . . . . . . . . . . . . . . 14
2.5.1 Bounding the Number of Monic Irreducible Polynomials . . . 17 2.5.2 Density of Monic Irreducible Polynomials . . . . . . . . . . . . 18 2.5.3 Aperiodic and Periodic Necklaces . . . . . . . . . . . . . . . . 19 v
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 An Analogue of the Prime Number Theorem for Polynomials over
Finite Fields

24

3.1 Enumeration of Prime Polynomials . . . . . . . . . . . . . . . . . . . . 25 3.2 Asymptotic Expansions of the Truncated Polylogarithm . . . . . . . . 29 3.3 The Prime Polynomial Theorem for Finite Fields . . . . . . . . . . . . 41 3.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 The T-Transform and T-Complexity

57

4.1 Background and Related Work . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Computational Complexity . . . . . . . . . . . . . . . . . . . . 57 4.1.2 Algorithmic Complexity . . . . . . . . . . . . . . . . . . . . . . 58 4.1.3 Deterministic Complexity and Randomness . . . . . . . . . . . 58
4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 T-Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 The T-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.1 The Naïve T-Transform Algorithm . . . . . . . . . . . . . . . . 66 4.4.2 T-Transform Algorithm Evolution . . . . . . . . . . . . . . . . . 69
4.5 T-Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5.1 Bounding T-Complexity . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 The T-Complexity of Uniformly Distributed Random Sequences

78

5.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Conjectures on the Statistics of the T-Transform . . . . . . . . . . . . . 79
5.2.1 The T-augmentation Level Distribution of Short Necklaces . . 82 5.2.2 The T-handle Length Distribution of Short Necklaces . . . . . 92 5.2.3 Beyond Short Necklaces . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Measuring String Similarity

104

6.1 The Normalized Information Distance . . . . . . . . . . . . . . . . . . 104 6.2 The Normalized Compression Distance . . . . . . . . . . . . . . . . . 105 vi
6.3 The Normalized T-Complexity Distance . . . . . . . . . . . . . . . . . 107
6.3.1 Metric Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7 Conclusions

114

7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A Supplemental Materials

117

A.1 Maple Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography

121

vii

List of Tables

Table 2.1 Table 2.2 Table 2.3
Finite ꢀeld representation for F2[t]/(t4 + t + 1). . . . . . . . . . 20 Cyclotomic cosets for F16. . . . . . . . . . . . . . . . . . . . . . 21

m

Cyclotomic cosets of F2 and binary necklaces of length m=4. 22
Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 3.5 Table 3.6 Table 3.7 Table 3.8
Asymptotic approximations to An,K(z) for z = 0.2. . . . . . . . 45 Asymptotic approximations to An,K(z) for z = 2. . . . . . . . . 46 Asymptotic approximations to An,K(z) for z = −7 + 11i. . . . . 47 Eulerian number triangle. . . . . . . . . . . . . . . . . . . . . . 48 Asymptotic approximations to LN (z,s,m) for z = 3, s = 1. . . . 49 Relative approximation error of LN (z,s,m) for z = 3, s = 1. . . . 51 Relative approximation error of LN (z,s,m) for z = 1.25, s = 2. . 52 Relative approximation error of LN (z,s,m) for z = −9 + 2.5i, s = 5. . . . . . . . . . . . . . . . . . . . . . . . . . 53

  • Absolute approximation error of the monic prime
  • Table 3.9

polynomial counting function for q = 2. . . . . . . . . . . . . . 54
Table 3.10 Relative approximation error of the monic prime polynomial counting function estimates for q = 2. . . . . . . . 55

  • Table 4.1
  • Computational complexity of LZ string factorization

algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Computational complexity of T-transform algorithms. . . . . . 70 Comparison of lower and asymptotic bound on maximal T-complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Comparison of upper and asymptotic bound on maximal T-complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Table 4.2 Table 4.3

Table 4.4

  • Table 5.1
  • Exponential T-augmentation level probability model

parameter estimation. . . . . . . . . . . . . . . . . . . . . . . . 87 viii

  • Table 5.2
  • Goodness of ꢀt test and exponential PDF parameter

estimations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Table 6.1 Table 6.2 Table 6.3
T-transform of string x#y. . . . . . . . . . . . . . . . . . . . . . 108 T-transform of string y#x. . . . . . . . . . . . . . . . . . . . . . 109 T-transform results for all pairwise concatenations of the three strings x, y, and z. . . . . . . . . . . . . . . . . . . . . . . 112 ix

List of Figures

Figure 2.1 Binary necklaces of length m = 4. . . . . . . . . . . . . . . . . . 22 Figure 3.1 Comparison of empirical and optimal truncation of
LN (z,s,m) for z = 3, s = 1. . . . . . . . . . . . . . . . . . . . . . 48
Figure 3.2 Absolute approximation error of LN (z,s,m) for z = 3, s = 1 under optimal truncation. . . . . . . . . . . . . . . 50

Figure 4.1 Example of a binary T-code construction. . . . . . . . . . . . . 64 Figure 4.2 Pseudo-code listing of naïve T-transform algorithm. . . . . . . 66 Figure 4.3 T-transform at intermediate T-augmentation level i. . . . . . . 67 Figure 4.4 Comparison of upper, lower, and asymptotic bound on maximal T-complexity. . . . . . . . . . . . . . . . . . . . . . . . 75

Figure 5.1 T-complexity of random sequence x versus minimal and maximal T-complexity bounds. . . . . . . . . . . . . . . . . . . 80
Figure 5.2 Histogram of υ(x) for |x| = 232 bits for 512 binary uniform random sequences. . . . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 5.3 Empirical and modelled cumulative distribution function of T-augmentation levels of x. . . . . . . . . . . . . . . 82
Figure 5.4 Probability of the occurrence of a necklace of length m at
T-augmentation level ℓ. . . . . . . . . . . . . . . . . . . . . . . . 84
Figure 5.5 Quantile-quantile plot for necklaces length 1 to 10 over ℓ. . . . 85 Figure 5.6 Quantile-quantile plot for necklaces length 11 to 20 over ℓ. . . 86 Figure 5.7 Empirical and modelled CDFs for T-augmentation level ℓ. . . 89 Figure 5.8 Error between empirical and modelled PDFs for
T-augmentation level ℓ with m from 1 to 10. . . . . . . . . . . . 90
Figure 5.9 Error between empirical and modelled PDFs for
T-augmentation level ℓ with m from 11 to 20. . . . . . . . . . . 91
Figure 5.10 Quantile-quantile plot for necklaces length 1 to 10 over h. . . . 93 x
Figure 5.11 Quantile-quantile plot for necklaces length 11 to 20 over h. . . 94 Figure 5.12 Empirical and modelled CDFs for T-handle length |x˜i|. . . . . . 95 Figure 5.13 Error between empirical and modelled PDFs for
T-handle length h with m from 1 to 10. . . . . . . . . . . . . . . 96
Figure 5.14 Error between empirical and modelled PDFs for
T-handle length h with m from 11 to 20. . . . . . . . . . . . . . 97
Figure 5.15 Modelled and average empirical necklace count per length m over 512 trials. . . . . . . . . . . . . . . . . . . . . . . 99
Figure 5.16 Sample standard deviation for the necklace count per length m over 512 trials. . . . . . . . . . . . . . . . . . . . . . . 100
Figure 5.17 Error between modelled and average empirical necklace count per length m. . . . . . . . . . . . . . . . . . . . . . . . . . 101
Figure 5.18 Logarithmic ratio of modelled and average empirical necklace count per length m. . . . . . . . . . . . . . . . . . . . . 102 xi

Nomenclature

Mathematical Functions

A(n,k)

An(z)

Bn

Eulerian number The nth Eulerian polynomial in z The nth Bernoulli number Bn = Bn(0) with B0 = 1 The nth Bernoulli polynomial in x T-complexity of the string x

Bn(x) CT (x) CT (x)

Maximal T-complexity bound of strings of length |x| Normalized information distance of x and y Normalized compression distance of x and y Normalized T-complexity distance of x and y Exponential integral

max

dNID(x,y)

dd

NC D(x,y) NT C(x,y)

Ei(x)

gcd(a,b)

ℑ(z)

Greatest common divisor of integers a and b Imaginary part of complex number z Least common multiple of integers a and b Logarithmic integral lcm(a,b)

li(x) Li(x)

Oꢁset logarithmic integral

Lis(z)

Polylogarithm, also known as Jonquière’s function Truncated polylogarithm function

L(z,s,m)

xii

  • logz
  • Natural logarithm function

logb z

Lp(m)

Logarithm function of base b The number of monic irreducible polynomials of degree m or the number of Lyndon words of length m

µ(n)

Möbius function

Nq(m)

The number of distinct monic irreducible polynomials over Fq of degree d 6 m such that d|m

O( · )

Landau gauge in asymptotics and big O notation in computer science

ord(a)

ϕ(n)

Order of the element a of a cyclic group Euler’s totient function

Φ(z,s,a) π(x)

Lerch transcendent Prime counting function enumerating the number of primes less than or equal to x

πq(m)

Prime polynomial counting function enumerating the monic irreducible polynomials of degree m or less in Fq[t]

PP(g,z) ℜ(z)

Principal part of the Laurent series of the function g about z Real part of complex number z

  • Res(g,z)
  • Residue of the function g at z

Mathematical Symbols and Notation

·

Operation or variable placeholder Set product or scalar multiplication Plus or minus

×

,

Negation of equality xiii

≡

∼

≪≫∀

Equality as per modulo operation mod Asymptoticity, f ∼ g implies f /g → 1 Much less than Much greater than For all

⊂∩∪\

Set containment relation Set intersection operator Set union operator Set diꢁerence

∅

Empty set

∈

Set membership

<

Negation of set membership

| · |

Magnitude of complex number, cardinality of set, or length of a string

:

{ }

Deꢀnes properties of elements of set Set of elements a, b, and c

{a,b,c} [a,b]

d|n

→

Set of real numbers between a and b The integer d divides the integer n Convergence

→

Function mapping

C

Set of complex numbers

Fq

Finite ꢀeld withq elements whereq is a prime power, Fq = (Zq,+,×) Multiplicative group of the ꢀnite ꢀeld Fq, Fq∗ = (Zq\{0},×)

Fq∗

xiv

Fq[t]

ki

Univariate polynomials with coeꢂcients in Fq The ith copy factor lim mod
Limit value Modulo operation, gives remainder after division of one integer by another

N

N+

N∗

pi

Set of nonnegative integers, N = {0,1,. . . } Set of positive integers excluding zero The index of least term of an optimally truncated series The ith copy pattern or the ith distinct prime factor of an integer Set of real numbers

R

R+

ρ

Set of positive real numbers excluding zero Signiꢀcance level of statistical test

S

Alphabet set S = {a1,a2,a3,. . . ,aq−1,aq} where ai are symbols Set of all strings including the empty string Set of all strings excluding the empty string T-code at T-augmentation level j

S∗

+

S

(k1,k2,...,kj )

S

(p1,p2,...,pj )

Z

Set of positive and negative integers including zero Set of positive integers excluding zero

Z+

Z−

Set of negative negative integers excluding zero Cyclic group of q elements

(Zq,·)

xv
ACKNOWLEDGEMENTS
The three years I have spent studying for this Ph.D. have been a great experience for me. I feel that I have learned a lot in this time, not only academically but also personally, and I am indebted to the many people that have supported me along the way.
First and foremost, I would like to thank my Co-supervisors Aaron Gulliver and Stephen Neville. Thank you Aaron for being so generous with your time, always asking the right questions, and motivating me to push further than I thought I could go. When I needed a little advice, you always had some to spare. Without your guidance I would not have written this dissertation. Your knowledge, enthusiasm, and sense of humour have been an inspiration for me and made my Ph.D. so enjoyable. Thank you Stephen for convincing me to do this Ph.D. in the ꢀrst place, your time, advice, feedback, and ꢀnancial support was much appreciated.
A very special thanks also goes to Ulrich Speidel at the University of Auckland, who has always been willing to share his advice and expertise with me. I loved going on all the hiking trips with you when you where here. Thanks for being such a great person and for reading all my drafts so carefully.
Thank you Dr. Wu-Sheng Lu for opening my eyes to convex optimization, you truly taught the best class I ever took.
Thanks also go to my many friends (you know who you are). Without you guys this Ph.D. would have been much less exciting.
A big thank-you also goes to my family away from home, Penny and Doug, thanks for being so wonderful. Most of all, I would like to thank my parents along with my brothers. Thank you for all your support and your continued love and encouragement over all these years. Papi, Mami, Jan, and Till thanks for always being there for me.

Divergent series are the invention of the devil, and it is shameful to base on them any demonstrations whatsoever.

Niels Henrik Abel xvi
DEDICATION
For my father, with love.
1

Chapter 1 Introduction

In mathematics, hardly any topic has intrigued curious minds more than the study of prime numbers. A prime number is a positive integer larger than one that has no positive divisors other than one and itself. All integers other than zero and one can be factored into a sequence of primes. However, for large integers prime factorization is a computationally hard problem which is exploited in cryptography for the secure exchange of information over an untrusted communication link.
Among the positive integers prime numbers seem to be randomly distributed, yet on close inspection their asymptotic distribution shows remarkable regularity. Towards the end of the 18th century Gauss noticed that the probability that a randomly chosen integer less than x is a prime number is close to 1/ logx. He later conjectured that the prime counting function enumerating the number of primes less than or equal to x is asymptotically given by the oꢁset logarithmic integral which may be approximated in terms of a divergent series expansion as follows

Recommended publications

  • In Combinatorics, the Eulerian Number A(N, M), Is the Number of Permutations of the Numbers 1 to N in Which Exactly M Elements A

    From Wikipedia, the free encyclopedia In combinatorics, the Eulerian number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). They are the coefficients of the Eulerian polynomials: The Eulerian polynomials are defined by the exponential generating function The Eulerian polynomials can be computed by the recurrence An equivalent way to write this definition is to set the Eulerian polynomials inductively by Other notations for A(n, m) are E(n, m) and . 1History 2 Basic properties 3 Explicit formula 4 Summation properties 5 Identities 6 Eulerian numbers of the second kind 7 References 8 External links In 1755 Leonhard Euler investigated in his book Institutiones calculi differentialis polynomials α1(x) = 1, α2(x) = x + 1, 2 α3(x) = x + 4x + 1, etc. (see the facsimile). These polynomials are a shifted form of what are now called the Eulerian polynomials An(x). For a given value of n > 0, the index m in A(n, m) can take values from 0 to n − 1. For fixed n there is a single permutation which has 0 ascents; this is the falling permutation (n, n − 1, n − 2, ..., 1). There is also a single permutation which has n − 1 ascents; this is the rising permutation (1, 2, 3, ..., n). Therefore A(n, 0) and A(n, n − 1) are 1 for all values of n. Reversing a permutation with m ascents creates another permutation in which there are n − m − 1 ascents. Therefore A(n, m) = A(n, n − m − 1).
  • A Unified Approach to Combinatorial Triangles: a Generalized Eulerian Polynomial

    A Unified Approach to Combinatorial Triangles: a Generalized Eulerian Polynomial

    A unified approach to combinatorial triangles: a generalized Eulerian polynomial Bao-Xuan Zhu∗ School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China Abstract Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array [ n,k]n,k 0 satisfying the recurrence relation: T ≥ cd n,k = λ(a0n + a1k + a2) n 1,k + (b0n + b1k + b2) n 1,k 1 + (n k + 1) n 1,k 2 T T − T − − λ − T − − with = 1 and = 0 unless 0 k n. We derive a functional transformation T0,0 Tn,k ≤ ≤ for its row-generating function (x) from the row-generating function A (x) of Tn n another array [An,k]n,k satisfying a two-term recurrence relation. Based on this transformation, we can get properties of and (x) including nonnegativity, Tn,k Tn log-concavity, real rootedness, explicit formula and so on. Then we extend the famous Frobenius formula, the γ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial. We also get an identity for the generalized Eulerian polynomial with the general derivative polynomial. Finally, we apply our results to an array from the Lambert function, a triangu- lar array from staircase tableaux and the alternating-runs triangle of type B in a unified approach. arXiv:2007.12602v1 [math.CO] 24 Jul 2020 MSC: 05A15; 05A19; 05A20; 26C10 Keywords: Eulerian numbers; Eulerian polynomials; Recurrence relations; Gener- ating functions; Real zeros; Log-concavity; Strong q-log-convexity; γ-positivity; 1 Introduction 1.1 Eulerian numbers Let Sn denote the symmetric group on n-elements [n]= 1, 2,...,n .
  • Factorial Stirling Matrix and Related Combinatorial Sequences

    Factorial Stirling Matrix and Related Combinatorial Sequences

    Linear Algebra and its Applications 357 (2002) 247–258 www.elsevier.com/locate/laa Factorial Stirling matrix and related combinatorial sequencesୋ Gi-Sang Cheon a,∗, Jin-Soo Kim b aDepartment of Mathematics, Daejin University, Pocheon 487-711, Republic of Korea bDepartment of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 21 November 2001; accepted 9 April 2002 Submitted by R.A. Brualdi Abstract In this paper, some relationships between the Stirling matrix, the Vandermonde matrix, the Benoulli numbers and the Eulerian numbers are studied from a matrix representation of k!S(n,k) which will be called the factorial Stirling matrix, where S(n,k) are the Stirling numbers of the second kind. As a consequence a number of interesting and useful identities are obtained. © 2002 Elsevier Science Inc. All rights reserved. AMS classification: 05A19; 05A10 Keywords: Stirling number; Bernoulli number; Eulerian number 1. Introduction In many contexts (see [1–4]), a number of interesting and useful identities involv- ing binomial coefficients are obtained from a matrix representation of a particular counting sequence. Such a matrix representation provides a powerful computational tool for deriving identities and an explicit formula related to the sequence. First, we begin our study with a well-known definition: for any real number x and an integer k,define ୋ This work was supported by the Daejin University Research Grants in 2001. ∗ Corresponding author. E-mail addresses: [email protected] (G.-S. Cheon), [email protected] (J.-S. Kim). 0024-3795/02/$ - see front matter 2002 Elsevier Science Inc. All rights reserved.
  • Bernoulli and Euler Polynomials

    Bernoulli and Euler Polynomials

    Bernoulli and Euler Polynomials Güneş Çatma Submitted to the Institute of Graduate Studies and Research in partial fulfillment of the requirements for the Degree of Master of Science in Mathematics Eastern Mediterranean University July 2013 Gazimağusa, North Cyprus Approval of the Institute of Graduate Studies and Research Prof. Dr. Elvan Yılmaz Director I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Mathematics. Prof. Dr. Nazım Mahmudov Chair, Department of Mathematics We certify that we have read this thesis and that in our opinion it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Mathematics. Assoc. Prof. Dr. Sonuç Zorlu Oğurlu Supervisor Examining Committee 1. Prof. Dr. Nazım Mahmudov 2. Assoc. Prof. Dr. Mehmet Ali Özarslan 3. Assoc. Prof. Dr. Sonuç Zorlu Oğurlu ABSTRACT This thesis provides an overview of Bernoulli and Euler numbers. It describes the Bernoulli and Euler polynomials and investigates the relationship between the classes of the two polynomials. It also discusses some important identities using the finite difference calculus and differentiation. The last part of this study is con- cerned with the Generalized Bernoulli and Euler polynomials. Furthermore, the properties obtained in the second chapter are also examined for the generalized Bernoulli and Euler polynomials in this part of the thesis. The Complemen- tary Argument Theorem, the generating functions, the Multiplication and the Euler-Maclauren Theorems are widely used in obtaining the mentioned results. Keywords: Bernoulli -Euler Polynomials, Generalized Bernoulli -Euler Polyno- mials, Finite Difference iii OZ¨ Bu ¸calı¸smadaBernoulli ve Euler sayıları ile Bernoulli ve Euler polinom- ları arasındaki ili¸skilerincelenmi¸stir.
  • Generalized Eulerian Numbers

    Generalized Eulerian Numbers

    JOURNAL OF COMBINATORIALTHEORY, Series A 32, 195-215 (1982) Generalized Eulerian Numbers D. H. LEHMER Department of Mathematics, University of Cal$ornia, Berkeley, California 94720 Received March 17, 1981 DEDICATED TO PROFESSOR MARSHALL HALL, JR., ON THE OCCASION OF HIS RETIREMENT 1. INTRODUCTION The Eulerian numbers (not to be confused with the Euler numbers E,) were discovered by Euler in 1755 [I]. Since then, they have been rediscovered, redefined, generalized and used by many authors [2-151. Explicitly they can be defined for 0 < k < n by k-l A(n,k)= 1 (-1)’ (k -j)“. j=O Euler wrote g+ f z-z,(x) P/n! n=O and (x - 1)” H,(x) = i A@, k) xk-‘. &=I This can be expressed by the generating function l-u = 1 + 2 f A@, k) ukt”/n! 1 - u exp{t(l - u)} &=I n=l from which the recurrence relation A@+ l,k)=(n-k+2)A(n,k- l)+kA(k,t) (3) is easily derived (see, for example, Comtet [ 15, Vol. 1, pp. 63-641. 195 0097-3 165/82/020195-21$02.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. 196 D. H. LEHMER The connections between A(n, k) and the polynomials of Bernoulli and Euler, such as ml(x)-B,(O)) = ;:r (X+;-l)A(n-l,k), were among the first discoveries made. Later Eulerian Numbers began to occur in combinatorial problems and A@, k) is best remembered as the number of permutations 1 2 )...) n $l), 742) ,***,70) that have precisely k - 1 “rises,” where a rise is a pair of consecutive marks n(i), n(i + 1) with n(i) < n(i + 1).
  • General Eulerian Polynomials As Moments Using Exponential Riordan Arrays

    General Eulerian Polynomials As Moments Using Exponential Riordan Arrays

    1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.9.6 47 6 23 11 General Eulerian Polynomials as Moments Using Exponential Riordan Arrays Paul Barry School of Science Waterford Institute of Technology Ireland [email protected] Abstract Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the general Eulerian polynomials, as defined by Xiong, Tsao and Hall, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of this polynomial sequence in terms of continued fractions, and we also calculate the Hankel transforms of the polynomial sequence. We indicate that the polynomial sequence can be characterized by the further notion of generalized Eulerian distribution first introduced by Morisita. We finish with examples of related Pascal-like triangles. 1 Introduction The triangle of Eulerian numbers 10 0 000 ... 10 0 000 ... 11 0 000 ... 14 1 000 ... , 1 11 11 1 0 0 ... 1 26 66 26 1 0 ... . . .. 1 with its general elements k n−k n +1 n +1 A = (−1)i (k − i + 1)n = (−1)i (n − k − i)n, n,k i i i=0 i=0 X X along with its variants, has been studied extensively [1, 10, 14, 15, 17, 18]. It is closely associated with the family of Eulerian polynomials k En(t)= An,kt . k X=0 The Eulerian polynomials have exponential generating function ∞ xn t − 1 E (t) = . n n! t − ex(t−1) n=0 X It can be shown that the sequence En(t) is the moment sequence of a family of orthogonal polynomials [3].
  • Eulerian Polynomials and B-Splines

    Eulerian Polynomials and B-Splines

    Eulerian Polynomials and B-Splines Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL 61702-2900, USA Abstract Here presented is the interrelationship between Eulerian polynomials, Eule- rian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, B- splines, respectively. The properties of Eulerian polynomials and Eulerian frac- tions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed. AMS Subject Classification: 41A80, 65B10, 33C45, 39A70. Key Words and Phrases: B-spline, Eulerian polynomial, Eulerian Number, Eulerian Fraction, exponential Euler polynomial, Euler-Frobenius polynomial, and Euler-Frobenius fractions. 1 Introduction Eulerian polynomial sequence fAn(z)gn≥0 is defined by the following summation (cf. for examples, [1], [2], and [3]): X An(z) `nz` = ; jzj < 1: (1.1) (1 − z)n+1 `≥0 It is well-known that the Eulerian polynomial, An(z) (see p. 244 of [3]), of degree n can be written in the form n X k An(z) = A(n; k)z ;A0(z) = 1; (1.2) k=1 where A(n; k) are called the Eulerian numbers that can be calculated by using k X n + 1 A(n; k) = (−1)j (k − j)n; 1 ≤ k ≤ n; (1.3) j j=0 1 2 T. X. He and A(n; 0) = 1. The Eulerian numbers are found, for example, in the standard treatises by Riordan [4], Comtet [3], Aigner [5], Grahm et al. [6], etc. However, their n notations are not standard, which are shown by A , A(n; k), W , , and n;k n;k k E(n; k), respectively.
  • Generalized J-Factorial Functions, Polynomials, and Applications

    Generalized J-Factorial Functions, Polynomials, and Applications

    1 2 Journal of Integer Sequences, Vol. 13 (2010), 3 Article 10.6.7 47 6 23 11 Generalized j-Factorial Functions, Polynomials, and Applications Maxie D. Schmidt University of Illinois, Urbana-Champaign Urbana, IL 61801 USA [email protected] Abstract The paper generalizes the traditional single factorial function to integer-valued mul- tiple factorial (j-factorial) forms. The generalized factorial functions are defined recur- sively as triangles of coefficients corresponding to the polynomial expansions of a subset of degenerate falling factorial functions. The resulting coefficient triangles are similar to the classical sets of Stirling numbers and satisfy many analogous finite-difference and enumerative properties as the well-known combinatorial triangles. The generalized triangles are also considered in terms of their relation to elementary symmetric poly- nomials and the resulting symmetric polynomial index transformations. The definition of the Stirling convolution polynomial sequence is generalized in order to enumerate the parametrized sets of j-factorial polynomials and to derive extended properties of the j-factorial function expansions. The generalized j-factorial polynomial sequences considered lead to applications expressing key forms of the j-factorial functions in terms of arbitrary partitions of the j-factorial function expansion triangle indices, including several identities related to the polynomial expansions of binomial coefficients. Additional applications include the formulation of closed-form identities and generating functions for the Stirling numbers of the first kind and r-order harmonic number sequences, as well as an extension of Stirling’s approximation for the single factorial function to approximate the more general j-factorial function forms. 1 Notational Conventions Donald E.
  • Generalized Euler Sums

    Generalized Euler Sums

    Generalized Eulerian Sums Fan Chung∗ Ron Graham∗ Dedicated to Adriano Garsia on the occasion of his 84th birthday. Abstract In this note, we derive a number of symmetrical sums involving Eu- lerian numbers and some of their generalizations. These extend earlier identities of Don Knuth and the authors, and also include several q-nomial sums inspired by recent work of Shareshian and Wachs on the joint distri- bution of various permutation statistics, such as the number of excedances, the major index and the number of fixed points of a permutation. We also produce symmetrical sums involving \restricted" Eulerian numbers which count permutations π on f1; 2; : : : ; ng with a given number of descents and which, in addition, have the value of π−1(n) specified. 1 Introduction n The classical Eulerian numbers, which we will denote by k , occur in a variety of contexts in combinatorics, number theory and computer science (see [4]). These numbers were introduced by Euler [3] in 1736 and have many interesting properties. We list some small values in Table 1. k 0 1 2 3 4 5 1 1 0 0 0 0 0 2 1 1 0 0 0 0 n 3 1 4 1 0 0 0 4 1 11 11 1 0 0 5 1 26 66 26 1 0 6 1 57 302 302 57 1 n Some Eulerian numbers k Table 1 n In particular, k enumerates the number of permutations π on [n] := f1; 2; : : : ; ng which have k descents (i.e., i ≤ n−1 with π(i) > π(i+1) ) as well as the number ∗University of California, San Diego 1 of permutations π on [n] which have k excedances (i.e., i < n with π(i) > i).
  • Eulerian Polynomials and Polynomial Congruences 1

    Eulerian Polynomials and Polynomial Congruences 1

    Volume 14, Number 1, Pages 46{54 ISSN 1715-0868 EULERIAN POLYNOMIALS AND POLYNOMIAL CONGRUENCES KAZUKI IIJIMA, KYOUHEI SASAKI, YUUKI TAKAHASHI, AND MASAHIKO YOSHINAGA Abstract. We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial. 1. Introduction The Eulerian polynomial A`(x)(` ≥ 1) was introduced by Euler in the study of sums of powers [5]. In this paper, we define the Eulerian polynomial A`(x) as the numerator of the rational function 1 ` X d 1 A`(x) (1.1) F (x) = k`xk = x = : ` dx 1 − x (1 − x)`+1 k=1 2 2 The first few examples are A1(x) = x; A2(x) = x + x ;A3(x) = x + 4x + 3 2 3 4 x ;A4(x) = x + 11x + 11x + x , etc. The Eulerian polynomial A`(x) is a monic of degree ` with positive integer coefficients. Write A`(x) = P` k k=1 A(`; k)x . The coefficient A(`; k) is called an Eulerian number. In the 1950s, Riordan [10] discovered a combinatorial interpretation of Eulerian numbers in terms of descents and ascents of permutations, and Carlitz [3] defined q-Eulerian numbers. Since then, Eulerian numbers are actively studied in enumerative combinatorics. (See [4, 11, 8].) Another combinatorial application of the Eulerian polynomial was found in the theory of hyperplane arrangements [15, 14]. The characteristic poly- nomial of the so-called Linial arrangement [9] can be expressed in terms of the root system generalization of Eulerian polynomials introduced by Lam and Postnikov [6]. The comparison of expressions in [9] and [15] yields the following.
  • Extensions of the Combinatorics of Poly-Bernoulli Numbers 3

    Extensions of the Combinatorics of Poly-Bernoulli Numbers 3

    EXTENSIONS OF THE COMBINATORICS OF POLY-BERNOULLI NUMBERS BEATA´ BENYI´ AND TOSHIKI MATSUSAKA Abstract. In this paper we extend the combinatorial theory of poly-Bernoulli numbers. We prove combinatorially some identities involving poly-Bernoulli polynomials. We in- troduce a combinatorial model for poly-Euler numbers and provide combinatorial proofs of some identities. We redefine r-Eulerian polynomials and verify Stephan’s conjecture concerning poly-Bernoulli numbers and the central binomial series. 1. Introduction Poly-Bernoulli numbers were introduced by Kaneko [14] using polylogarithm function. His work motivated several studies in the same manner, such as further generalizations of these numbers and analogue definitions based on the polylogarithm function. These numbers and their relatives have also beautiful combinatorial properties and arise in dif- ferent research fields, such as discrete tomographie, mathematics of origami, biology, etc. [13, 25, 27, 34]. In this work we show some possible directions of studies motivated by the combinatorial treatment of these numbers. First, we introduce two models for interpreting the poly- Bernoulli polynomials and in order to show their usefulness we provide combinatorial proofs for some known identities. In Section 3 we consider poly-Euler numbers and extend one of the combinatorial interpretations of poly-Bernoulli numbers that helps in the combinatorial analysis of poly-Euler numbers of the first and second kind studied in [20, 32]. Section 4 is devoted to r-Eulerian polynomials, which we define a new way and use for verifying a conjecture concerning a relation between the central binomial series and poly-Bernoulli numbers observed by Stephan and stated by Kaneko [15].
  • Permutations, Products, Posets, and Pfaffians

    Permutations, Products, Posets, and Pfaffians

    University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2015 Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians Norman B. Fox University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Fox, Norman B., "Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians" (2015). Theses and Dissertations--Mathematics. 25. https://uknowledge.uky.edu/math_etds/25 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work.