Gems in experimental mathematics: AMS Special Session on Experimental Mathematics, January 5, 2009, Washington, D.C

These proceedings reflect the special session on Experimental Mathematics held January 5, 2009, at the Joint Mathematics Meetings in Washington, DC as well as some papers specially solicited for this volume. Experimental Mathematics is a recently structured field of Mathematics that uses the computer and advanced computing technology as a tool to perform experiments. These include the analysis of examples, testing of new ideas, and the search of patterns to suggest results and to complement existing analytical rigor. The development of a broad spectrum of mathematical software products, such as Mathematica® and MapleTM, has allowed mathematicians of diverse backgrounds and interests to use the computer as an essential tool as part of their daily work environment. This volume reflects a wide range of topics related to the young field of Experimental Mathematics. The use of computation varies from aiming to exclude human input in the solution of a problem to traditional mathematical questions for which computation is a prominent tool Contents 6 Preface 8 The art of finding Calabi-Yau differential equations. Dedicated to the 90-th birthday of Lars Gårding 10 A note on a question due to A. Garsia 28 Experimental computation with oscillatory integrals 34 Experimental mathematics and mathematical physics 50 An extension of the parallel Risch algorithm 68 Appell polynomials and their zero attractors 78 Congruences for Stirling numbers of the second kind 106 Expressions for harmonic number exponential generating functions 122 Theory of log-rational integrals 136 A new algorithm for the recursion of hypergeometric multisums with improved universal denominator 152 1. Introduction 152 2. Two algorithms 154 3. Use of the program and examples 156 4. Proof of the multisum algorithm 157 5. Proof of the universal denominator algorithm 161 References 165 The method of brackets. Part 2: Examples and applications 166 History of the formulas and algorithms for π 182 A matrix form of Ramanujan-type series for 1/π 198 An algorithmic approach to the Mellin transform method 216 Eliminating human insight: An algorithmic proof of Stembridge’s TSPP theorem 228 Towards the Koch snowflake fractal billiard: Computer experiments and mathematical conjectures 240 An experimental mathematics perspective on the old, and still open, question of when to stop? 274 The distance to an irreducible polynomial 284 Square roots of 2 × 2 matrices 298 On a series of Ramanujan 314 Finite analogs of Szemerédi’s theorem 322 Towards an automation of the circle method 330 The greatest common divisor of an − 1 and bn − 1 and the Ailon-Rudnick conjecture 348 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 358 Experimentation at the frontiers of reality in Schubert calculus 374 On Sp4 modularity of Picard-Fuchs differential equations for Calabi-Yau threefolds 390 "These proceedings reflect the special session on Experimental Mathematics held January 5, 2009, at the Joint Mathematics Meetings in Washington, DC as well as some papers specially solicited for this volume." "Experimental Mathematics is a recently structured field of Mathematics that uses the computer and advanced computing technology as a tool to perform experiments. These include the analysis of examples, testing of new ideas, and the search of patterns to suggest results and to complement existing analytical rigor." "The development of a broad spectrum of mathematical software products, such as Mathematica® and Maple & trade; has allowed mathematicians of diverse backgrounds and interests to use the computer as an essential tool as part of their daily work environment." "This volume reflects a wide range of topics related to the young field of Experimental Mathematics. The use of computation varies from aiming to exclude human input in the solution of a problem to traditional mathematical questions for which computation is a prominent tool."--Jacket. The essays in this volume examine a wide range of topics related to the young field of experimental mathematics, and are based around the proceedings of the special session on Experimental Mathematics held in 2009, at the Joint Mathematics Meetings in Washington, DC.