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Gems in Experimental Mathematics PDF

426 Pages·2010·5.901 MB·English
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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 Gems in Experimental Mathematics This page intentionally left blank 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 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2010bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 151413121110 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 This page intentionally left blank 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) 11

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