Arithmetic aspects of random walks and methods in definite integration

421 Pages·2012·3.65 MB·English

Arithmetic aspects of random walks
and methods in definite integration
An Abstract
Submitted On The First Day Of May, 2012
To The Department Of Mathematics
In Partial Fulfillment Of The Requirements
Of The School Of Science And Engineering
Of Tulane University
For The Degree Of
Doctor Of Philosophy
By
Armin Straub
Approved:
Victor H. Moll, Ph.D.
Director
Tewodros Amdeberhan, Ph.D.
Jonathan M. Borwein, D.Phil.
Mahir B. Can, Ph.D.
Morris Kalka, Ph.D.
Abstract
Inthefirstpartofthisthesis, werevisitaclassicalproblem: howfardoesarandom
walk travel in a given number of steps (of length 1, each taken along a uniformly
random direction)? We study the moments of the distribution of these distances
as well as the corresponding probability distributions. Although such random walks
are asymptotically well understood, very few exact formulas had been known; we
supplement these with explicit hypergeometric forms and unearth general structures.
Our investigation of the moments naturally leads us to consider (multiple) Mahler
measures. For several families of Mahler measures we are able to give evaluations in
terms of log-sine integrals. Therefore, and because of the connections of log-sine
integrals with number theory and mathematical physics, we study generalized log-
sineintegralsandshowthattheyevaluateintermsofthewell-studiedpolylogarithms.
A computer algebra implementation of our results demonstrates that a large body of
results on log-sine integrals scattered over the literature is now computer-amenable.
The second part is concerned with developing specific methods for evaluating
several families of definite integrals arising in diverse contexts (such as calculations in
quantum field theories). We also review and illustrate Ramanujan’s Master Theorem
and show that it generalizes to the method of brackets, which has its roots in the
negative dimensional integration method utilized by particle physicists. We then
apply this technique to multiple integrals recently studied in a physical context.
Complementary to these symbolic methods, we present an exponentially fast algo-
rithm for numerically integrating rational functions over the real line. This algorithm
operates on the coefficients of the rational function instead of evaluating it.
Arithmetic aspects of random walks
and methods in definite integration
A Dissertation
Submitted On The First Day Of May, 2012
To The Department Of Mathematics
In Partial Fulfillment Of The Requirements
Of The School Of Science And Engineering
Of Tulane University
For The Degree Of
Doctor Of Philosophy
By
Armin Straub
Approved:
Victor H. Moll, Ph.D.
Director
Tewodros Amdeberhan, Ph.D.
Jonathan M. Borwein, D.Phil.
Mahir B. Can, Ph.D.
Morris Kalka, Ph.D.

Acknowlegdment
Let me begin by thanking my collaborators and coauthors who have directly con-
tributed to the work presented in this thesis:
Tewodros Amdeberhan (Tulane University),
•
David Borwein (University of Western Ontario),
•
Jonathan M. Borwein (University of Newcastle, Australia),
•
Olivier Espinosa (then Universidad T´ecnica Federico Santa Mar´ıa, Chile),
•
Ivan Gonzalez (Universidad T´ecnica Federico Santa Mar´ıa, Chile),
•
Marshall Harrison (Prestadigital LLC, Texas),
•
Dante Manna (then Virginia Wesleyan College),
•
Luis Medina (then Rutgers University),
•
Victor H. Moll (Tulane University),
•
Dirk Nuyens (K.U.Leuven, Belgium),
•
James Wan (University of Newcastle, Australia), and
•
Wadim Zudilin (University of Newcastle, Australia).
•
You have shown me that the beautiful and exciting land of mathematics is most
enjoyably explored in company and with guidance. All the papers presented in this
thesis, with the exception of the ones in Chapters 15 and 16, have been worked out
with at least one coauthor.
I am particularly grateful to my advisor Victor Moll who has always been an
unfailing source of energy and support. Victor is full of ideas and advice, and has
the gift of getting you hooked on interesting problems which he naturally attracts.
ii
Also, his ability to remember and come up with relevant references is remarkable and
has been very useful to me. Victor is a pleasure to work with and the main reason I
returned to Tulane to pursue a doctoral degree.
Tewodros Amdeberhan is one of the most enthusiastic and friendly persons I
have met. His enormous enthusiasm, encouragement and curiosity have supported
me throughout my times at Tulane. Early on, Teddy engaged me in mathematical
research and my first publication [Str08], presented in Chapter 15, is owed to his
encouragement. I am much indebted for his support, occasionally shared over a cup
of coffee, as a mentor and friend.
It was most fortunate for me, both mathematically and personally, to have had
the opportunity to visit Jonathan Borwein at Newcastle University. His industrious
and resourceful character continue to have a contagious effect on me, which makes
our collaboration not only fruitful but very enjoyable. I also thank Jon and his wife
Judith for several cultural and culinary introductions to Australia.
At Newcastle University, I have further benefited considerably from the presence
of James Wan and Wadim Zudilin. James, a fellow graduate student, is full of in-
triguing thoughts, both mathematical and philosophical. I have enjoyed his company
immensely and am fond to recall our excursions in Newcastle, Vancouver and San
Francisco. I am very thankful to Wadim, a true scholar, for sharing his remarkable
mathematical knowledge on many occasions. His interest and his friendly support
have been a delight.
I also wish to thank Manuel Kauers and Christoph Koutschan from RISC for sev-
eralhelpfuldiscussionsonapplicationsofcomputeralgebra. Inparticular,Christoph’s
package HolonomicFunctions, [Kou10], has been most helpful as has been his advice
on using it.
More specifically, the work in Chapter 2 benefited from helpful suggestions by
David Bailey, David Broadhurst and Richard Crandall. Further thanks are due to
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Bruno Salvy for reminding us of the existence of [Bar64], Michael Mossinghoff for
showing us [Klu06], and to Peter Donovan for stimulating the research on random
walks and for much subsequent useful discussion. We are grateful to Wadim Zudilin
formuchusefuldiscussionontheworkinChapter3,aswellasforpointingout[Bai32],
[Nes03],and[Zud02],whichhavebeencrucialinobtainingtheclosedformsofW ( 1).
4
±
We thank Michael Mossinghoff for pointing us to the Mahler measure conjectures via
[RVTV04], and Plamen Djakov and Boris Mityagin for correspondence related to
Theorem 4.2.7 and the history of their proof. We are specially grateful to Don Zagier
for not only providing us with his proof of Theorem 4.2.7 but also for his enormous
amount of helpful comments which improved Chapter 4. We are much obliged to
David Bailey for significant numerical assistance in Chapters 4 and 7, especially with
the two-dimensional quadratures in Example 7.9.2. Thanks are due to Yasuo Ohno
and Yoshitaka Sasaki for introducing us to the relevant papers introducing multiple
and higher Mahler measure during a visit to CARMA. This initiated the work in
Chapters 6 and 7. Chapter 6 benefited from many useful discussions with James Wan
and David Borwein. We are grateful to Andrei Davydychev and Mikhail Kalmykov
for several valuable comments on an earlier version of work presented in Chapter 5
and for pointing us to relevant publications. We also wish to thank Bruce Berndt
for pointing out the results described in Section 12.8 of Chapter 12. Chapter 13 was
improved by Larry Glasser who pointed us to the sinc integral problem after hearing
a lecture on [BB01] and who provided historic context. We are also thankful for his
and Tewodros Amdeberhan’s comments on an earlier draft of the work in Chapter 13.
The topic of Chapter 15 was suggested by Tewodros Amdeberhan and I am especially
thankful for his continuous and helpful support on this subject. I also appreciate the
helpreceivedbyDoronZeilbergerwhosharedhisexpertiseintheformofusefulMaple
routines. Most parts of Chapter 16 have been written while I had the pleasure to
visit Marc Chamberland at Grinnell College. I am thankful for his encouraging and
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helpful support.
Throughout my time as a graduate student I have received gracious financial
support from various sources. As a graduate student, I was funded by teaching assis-
tantships, fellowship awards and tuition scholarships awarded by Tulane University
for the years 2006/2007 and 2008–2012. As a student of Victor Moll, I was also
supported by the NSF grants NSF-DMS 0070567, NSF-DMS 0070757, NSF-DMS
0409968, and NSF-DMS 0713836. During the academic year 2009/2010 I received
support as an IBM Fellow in Computational Science. The three visits at Newcastle
University were generously funded by Jonathan Borwein with support from the Aus-
tralian Research Council. Attending the conferences FPSAC 2009, 2010, 2011 as well
as the AMS Joint Meetings 2012 was made possible by kind travel support from the
NSF.
Finally, I wish to thank Victor Moll, Tewodros Amdeberhan, Jonathan Borwein,
Mahir Can and Morris Kalka for serving on my dissertation committee.
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Table of Contents
Acknowledgement ii
1 Introduction and overview 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Arithmetic aspects of random walks . . . . . . . . . . . . . . . . . 4
1.3 Methods in definite integration . . . . . . . . . . . . . . . . . . . . 7
1.4 Some notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Some arithmetic properties of short random walk integrals 12
2.1 Introduction, history and preliminaries . . . . . . . . . . . . . . . 13
2.2 The even moments and their combinatorial features . . . . . . . . 18
2.3 Analytic features of the moments . . . . . . . . . . . . . . . . . . 25
2.4 Bessel integral representations . . . . . . . . . . . . . . . . . . . . 31
2.5 The odd moments of a three-step walk . . . . . . . . . . . . . . . 32
2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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3 Three-step and four-step random walk integrals 43
3.1 Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . 43
3.2 Bessel integral representations . . . . . . . . . . . . . . . . . . . . 46
3.3 Probabilistically inspired representations . . . . . . . . . . . . . . 60
3.4 Partial resolution of Conjecture 3.1.1 . . . . . . . . . . . . . . . . 69
4 Densities of short uniform random walks 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 The densities p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
n
4.3 The density p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3
4.4 The density p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4
4.5 The density p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5
4.6 Derivative evaluations of W . . . . . . . . . . . . . . . . . . . . . 97
n
4.7 New results on the moments W . . . . . . . . . . . . . . . . . . . 104
n
4.8 Appendix: A family of combinatorial identities . . . . . . . . . . . 110
5 Special values of generalized log-sine integrals 114
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Evaluations at π . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3 Quasiperiodic properties . . . . . . . . . . . . . . . . . . . . . . . 125
5.4 Evaluations at other values . . . . . . . . . . . . . . . . . . . . . . 127
5.5 Reducing polylogarithms . . . . . . . . . . . . . . . . . . . . . . . 134
5.6 The program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
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