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This book presents a printed testimony for the fact that George Andrews, one of the world’s leading experts in partitions and q-series for the last several decades, has passed the milestone age of 80. To honor George Andrews on this occasion, the conference “Combinatory Analysis 2018” was organized at the Pennsylvania State University from June 21 to 24, 2018. This volume comprises the original articles from the Special Issue “Combinatory Analysis 2018 – In Honor of George Andrews’ 80th Birthday” resulting from the conference and published in Annals of Combinatorics. In addition to the 37 articles of the Andrews 80 Special Issue, the book includes two new papers. These research contributions explore new grounds and present new achievements, research trends, and problems in the area. The volume is complemented by three special personal contributions: “The Worlds of George Andrews, a daughter’s take” by Amy Alznauer, “My association and collaboration with George Andrews” by Krishna Alladi, and “Ramanujan, his Lost Notebook, its importance” by Bruce Berndt. Another aspect which gives this Andrews volume a truly unique character is the “Photos” collection. In addition to pictures taken at “Combinatory Analysis 2018”, the editors selected a variety of photos, many of them not available elsewhere: “Andrews in Austria”, “Andrews in China”, “Andrews in Florida”, “Andrews in Illinois”, and “Andrews in India”. This volume will be of interest to researchers, PhD students, and interested practitioners working in the area of Combinatory Analysis, q-Series, and related fields.

Number theory. --- Special functions. --- Combinatorics. --- Number Theory. --- Special Functions. --- Combinatorics --- Algebra --- Mathematical analysis --- Special functions --- Number study --- Numbers, Theory of --- Combinatorial analysis

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This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.

q-series. --- Mathematics. --- Ramanujan Aiyangar, Srinivasa, --- Math --- Science --- Series --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Geometry, algebraic. --- Sequences (Mathematics). --- Functions, special. --- Algebraic Geometry. --- Sequences, Series, Summability. --- Special Functions. --- Special functions --- Mathematical analysis --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Special functions.

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In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook.  In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series.  Most of the entries examined in this volume fall under the purviews of number theory and classical analysis.  Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed.  Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysis and prime number theory.   Most of the entries in number theory fall under the umbrella of classical analytic number theory.   Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society.

Mathematical analysis. --- Mathematicians -- India -- Biography. --- Number theory. --- Ramanujan Aiyangar, Srinivasa, 1887-1920. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Mathematicians --- Ramanujan Aiyangar, Srinivasa, --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Mathematics. --- Analysis (Mathematics). --- Fourier analysis. --- Special functions. --- Number Theory. --- Analysis. --- Fourier Analysis. --- Special Functions. --- Global analysis (Mathematics). --- Functions, special. --- Special functions --- Mathematical analysis --- Analysis, Fourier --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Number study --- Numbers, Theory of --- 517.1 Mathematical analysis

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In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This fifth and final installment of the authors’ examination of Ramanujan’s lost notebook focuses on the mock theta functions first introduced in Ramanujan’s famous Last Letter. This volume proves all of the assertions about mock theta functions in the lost notebook and in the Last Letter, particularly the celebrated mock theta conjectures. Other topics feature Ramanujan’s many elegant Euler products and the remaining entries on continued fractions not discussed in the preceding volumes. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews and Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society.

Mathematical analysis. --- Mathematicians --- 517.1 Mathematical analysis --- Mathematical analysis --- Ramanujan Aiyangar, Srinivasa, --- Ṣrīnivāsa-Rāmānuja Aiyaṅgār, --- Ramanujan, Srinivasa, --- Aiyangar, Srinivasa Ramanujan, --- Iyengar, Srinivasa Iyengar Ramanuja, --- Ramanuja Iyengar, Srinivasa Iyengar, --- Ramanudzhan Aĭengar, --- Ramanujan, S. --- Ramanujam, S. --- Functions, special. --- Functions of complex variables. --- Number theory. --- Special Functions. --- Functions of a Complex Variable. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Complex variables --- Elliptic functions --- Functions of real variables --- Special functions --- Special functions.

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Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematical physics, differential equations, mathematical computing, number theory, and combinatorics.

Functions, Special. --- Special functions --- Mathematical analysis --- Functions, Special --- Fonctions spéciales --- Hypergeometric functions. --- Hypergeometric series.