Gems in Experimental Mathematics

426 Pages·2010·5.901 MB·English

by  Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll, Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll (ed.)

C
ONTEMPORARY
M
ATHEMATICS
517
Gems in Experimental
Mathematics
AMS Special Session
Experimental Mathematics
January 5, 2009
Washington, DC
Tewodros Amdeberhan
Luis A. Medina
Victor H. Moll
Editors
American Mathematical Society
Gems in Experimental
Mathematics
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C
ONTEMPORARY
M
ATHEMATICS
517
Gems in Experimental
Mathematics
AMS Special Session
Experimental Mathematics
January 5, 2009
Washington, DC
Tewodros Amdeberhan
Luis A. Medina
Victor H. Moll
Editors
American Mathematical Society
Providence, Rhode Island
Editorial Board
Dennis DeTurck, managing editor
George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary 05A17, 11A05, 11A41, 11C08, 11F46,
11P55, 11Y60, 14J32, 14N15, 15A24, 37D40, 37D50, 65D18, 68R05.
Library of Congress Cataloging-in-Publication Data
AMSSpecialSessiononExperimentalMathematics(2009: Washington,D.C.)
Gems in experimental mathematics : AMS Special Session on Experimental Mathematics,
January 5, 2009, Washington, D.C. / Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll,
editors.
p.cm. —(Contemporarymathematics;v.517)
Includesbibliographicalreferences.
ISBN978-0-8218-4869-2(alk.paper)
1. Combinatorial analysis—Congresses. 2. Number theory—Congresses. 3. Experimental
mathematics—Congresses. I. Amdeberhan, Tewodros, 1971– II. Medina, Luis A., 1981–
III.Moll,VictorH.,1956– IV.Title.
QA164.A475 2009
511(cid:2).6—dc22
2010006127
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Contents
Preface vii
The art of finding Calabi-Yau differential equations. Dedicated to the 90-th
birthday of Lars G˚arding
Gert Almkvist 1
A note on a question due to A. Garsia
Tewodros Amdeberhan 19
Experimental computation with oscillatory integrals
David H. Bailey and Jonathan M. Borwein 25
Experimental mathematics and mathematical physics
David H. Bailey, Jonathan M. Borwein, David Broadhurst and
Wadim Zudilin 41
An extension of the parallel Risch algorithm
Stefan T. Boettner 59
Appell polynomials and their zero attractors
Robert P. Boyer and William M. Y. Goh 69
Congruences for Stirling numbers of the second kind
O-Yeat Chan and Dante Manna 97
Expressions for harmonic number exponential generating functions
Mark W. Coffey 113
Theory of log-rational integrals
Richard E. Crandall 127
A new algorithm for the recursion of hypergeometric multisums with improved
universal denominator
Stavros Garoufalidis and Xinyu Sun 143
The method of brackets. Part 2: Examples and applications
Ivan Gonzalez, Victor H. Moll and Armin Straub 157
History of the formulas and algorithms for π
Jesu´s Guillera 173
A matrix form of Ramanujan-type series for 1/π
Jesu´s Guillera 189
v
vi CONTENTS
An algorithmic approach to the Mellin transform method
Karen Kohl and Flavia Stan 207
Eliminating human insight: An algorithmic proof of Stembridge’s TSPP
theorem
Christoph Koutschan 219
Towards the Koch snowflake fractal billiard: Computer experiments and
mathematical conjectures
Michel L. Lapidus and Robert G. Niemeyer 231
An experimental mathematics perspective on the old, and still open, question
of when to stop?
Luis A. Medina and Doron Zeilberger 265
The distance to an irreducible polynomial
Michael J. Mossinghoff 275
Square roots of 2×2 matrices
Sam Northshield 289
On a series of Ramanujan
Olivier Oloa 305
Finite analogs of Szemer´edi’s theorem
Paul Raff and Doron Zeilberger 313
Towards an automation of the circle method
Andrew V. Sills 321
The greatest common divisor of an−1 and bn−1 and the Ailon-Rudnick
conjecture
Joseph H. Silverman 339
Which partial sums of the Taylor series for e are convergents to e? (and a link
to the primes 2, 5, 13, 37, 463). Part II
Jonathan Sondow and Kyle Schalm 349
Experimentation at the frontiers of reality in Schubert calculus
Christopher Hillar, Luis Garc´ıa-Puente, Abraham Mart´ın del
Campo, James Ruffo, Zach Teitler, Stephen L. Johnson, and Frank
Sottile 365
On Sp modularity of Picard-Fuchs differential equations for Calabi-Yau
4
threefolds
Yifan Yang and Wadim Zudilim 381
Preface
The editors of these proceedings organized a special session on Experimental
MathematicsattheJointMeetingsoftheAmericanMathematicalSocietythatwas
held in Washington, D. C., January 2009.
The point of view of Experimental Mathematics has been recently formalized
by a small number of research groups. The basic philosophical principles of this
branch of mathematics have appeared in books initiated by the groups around
David Bailey, Jonathan Borwein, Doron Zeilberger among others.
Thegoaloftheeditorsistobringtothisvolumeacollectionofpapersreflecting
the experimental nature of many mathematical problems. In a real sense, this
collection is a continuation of Tapas in Experimental Mathematics, volume 457
of this series.
ThevolumecontainsmostofthelecturespresentedattheWashingtonmeeting,
as well as some papers specially requested by the editors. An effort was made to
includeauthorsnottraditionallyincludedintheExperimentalMathematicsworld.
The editors wish to thank all participants at the special session and all con-
tributors to this volume. We also wish to acknowledge the invaluable help of the
referees and the American Mathematical Society, in particular the editors of the
ContemporaryMathematicsseries,thathavemadethisvolumeareality. Itishoped
that the papers appearing here will inspire many researchers to join the growing
Experimental Mathematics community.
The editors
New Orleans and San Juan
January 2010
vii
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CCoonntteemmppoorraarryyMMaatthheemmaattiiccss
Volume517,2010
The art of finding Calabi-Yau differential equations
Dedicated to the 90-th birthday of Lars G˚arding.
Gert Almkvist
Abstract. In this paper various methods for finding Calabi-Yau differential
equationsarediscussed. Theyareformalizedversionsofthedifferentialequa-
tions satisfied by the periods of Calabi-Yau manifolds in String Theory but
occuraswellinotherpartsofphysics.
1. Introduction
In 2003 van Straten and van Enckevort during a computer search found the
third order differential operator (η)
L=θ3−x(2θ+1)(11θ2+11θ+5)+125x2(θ+1)3.
Here θ =x d . Then Ly =0 where
dx
(cid:2)∞
y = A xn
n
n=0
for some unknown coefficients A , n = 0,1,2,... It took me five years before I
n
found an explicit expression for A , namely A =1 and
n 0
(cid:3) (cid:4) (cid:3) (cid:4) (cid:3) (cid:4) (cid:3) (cid:4)(cid:3) (cid:4)
2n −1 3n −1[(cid:2)n/5] n−2k n −2 n 5n−5k −1(5k)!(5n−5k)!
A =5 (−1)k
n n n 4n−5k k 5k n k!5 (n−k)!5
k=0
for n>0.How is it possible tofindsuch acomplicatedformula? Letus first point
out that I immediately got an e-mail from Zudilin where he (trivially) simplified
the formula to
(cid:3) (cid:4) (cid:3) (cid:4)
[(cid:2)n/5] n−2k n 3 4n−5k
A =5 (−1)k
n 4n−5k k 3n
k=0
and later (not so trivially) to
(cid:3) (cid:4) (cid:5)(cid:3) (cid:4) (cid:3) (cid:4)(cid:6)
[(cid:2)n/5] n 3 4n−5k−1 4n−5k
A = (−1)k + .
n k 3n 3n
k=0
Here it is evident that A is an integer. But it was found in the complicated form
n
above and here we shall tell the long story how I came to let Maple’s Zeilberger
find the recursion formula for this monster.
(cid:2)c2010 Americ(cid:2)acn00M00at(hceompyartiigcahltShoolcdieetry)
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