Download E-books Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions (Developments in Mathematics) PDF

The challenge of representing an integer as a sum of squares of integers is without doubt one of the oldest and most vital in arithmetic. It is going again at the least 2000 years to Diophantus, and maintains extra lately with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic functionality procedure dates from his epic Fundamenta Nova of 1829. right here, the writer employs his combinatorial/elliptic functionality how you can derive many countless households of specific certain formulation regarding both squares or triangular numbers, of which generalize Jacobi's (1829) four and eight squares identities to 4n² or 4n(n+1) squares, respectively, with no utilizing cusp types resembling these of Glaisher or Ramanujan for sixteen and 24 squares. those effects rely on new expansions for powers of varied items of classical theta features. this can be the 1st time that limitless households of non-trivial targeted particular formulation for sums of squares were chanced on.

The writer derives his formulation through the use of combinatorics to mix quite a few tools and observations from the idea of Jacobi elliptic services, persevered fractions, Hankel or Turanian determinants, Lie algebras, Schur features, and a number of simple hypergeometric sequence relating to the classical teams. His effects (in Theorem 5.19) generalize to split limitless households all the 21 of Jacobi's explicitly said measure 2, four, 6, eight Lambert sequence expansions of classical theta capabilities in sections 40-42 of the Fundamental Nova. the writer additionally makes use of a unique case of his tips on how to provide a derivation evidence of the 2 Kac and Wakimoto (1994) conjectured identities touching on representations of a favorable integer by way of sums of 4n² or 4n(n+1) triangular numbers, respectively. those conjectures arose within the examine of Lie algebras and feature additionally lately been proved by way of Zagier utilizing modular kinds. George Andrews says in a preface of this booklet, `This notable paintings will surely spur others either in elliptic features and in modular kinds to construct on those amazing discoveries.'

Audience: This learn monograph on sums of squares is extraordinary via its range of equipment and wide bibliography. It includes either targeted proofs and diverse particular examples of the idea. This readable paintings will entice either scholars and researchers in quantity conception, combinatorics, unique services, classical research, approximation thought, and mathematical physics.