The challenge of representing an integer as a sum of squares of integers is likely one of the oldest and most important in arithmetic. It is going again a minimum of 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 method dates from his epic Fundamenta Nova of 1829. the following, the writer employs his combinatorial/elliptic functionality how to derive many limitless households of particular targeted 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 varieties 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 precise particular formulation for sums of squares were came across.

The writer derives his formulation by using combinatorics to mix quite a few equipment and observations from the idea of Jacobi elliptic services, persisted fractions, Hankel or Turanian determinants, Lie algebras, Schur features, and a number of uncomplicated hypergeometric sequence with regards to the classical teams. His effects (in Theorem 5.19) generalize to split countless households all the 21 of Jacobi's explicitly acknowledged measure 2, four, 6, eight Lambert sequence expansions of classical theta services in sections 40-42 of the Fundamental Nova. the writer additionally makes use of a distinct case of his the way to provide a derivation evidence of the 2 Kac and Wakimoto (1994) conjectured identities referring to representations of a favorable integer by way of sums of 4n2 or 4n(n+1) triangular numbers, respectively. those conjectures arose within the research of Lie algebras and feature additionally lately been proved via Zagier utilizing modular types. George Andrews says in a preface of this publication, `This remarkable paintings will surely spur others either in elliptic capabilities and in modular kinds to construct on those exceptional discoveries.'

Audience: This learn monograph on sums of squares is unique by means of its variety of equipment and large bibliography. It includes either specific proofs and diverse specific examples of the speculation. This readable paintings will entice either scholars and researchers in quantity thought, combinatorics, designated capabilities, classical research, approximation concept, and mathematical physics.

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