By Stephen C. Milne

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 strategy dates from his epic **Fundamenta Nova** of 1829. the following, the writer employs his combinatorial/elliptic functionality ways to derive many countless households of particular special formulation related to both squares or triangular numbers, of which generalize Jacobi's (1829) four and eight squares identities to 4*n*^{2} or 4*n*(*n*+1) squares, respectively, with out utilizing cusp types comparable to these of Glaisher or Ramanujan for sixteen and 24 squares. those effects depend on new expansions for powers of assorted items of classical theta services. this can be the 1st time that limitless households of non-trivial distinctive specific formulation for sums of squares were discovered.

The writer derives his formulation through the use of combinatorics to mix quite a few tools and observations from the idea of Jacobi elliptic capabilities, persevered 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 endless 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 different case of his tips on how to supply a derivation evidence of the 2 Kac and Wakimoto (1994) conjectured identities bearing on representations of a favorable integer by means of sums of 4*n*^{2} or 4*n*(*n*+1) triangular numbers, respectively. those conjectures arose within the research of Lie algebras and feature additionally lately been proved through Zagier utilizing modular types. George Andrews says in a preface of this e-book, `This striking paintings will unquestionably spur others either in elliptic capabilities and in modular kinds to construct on those superb discoveries.'

*Audience:* This learn monograph on sums of squares is individual through its variety of tools and wide bibliography. It comprises either particular proofs and various particular examples of the idea. This readable paintings will attract either scholars and researchers in quantity thought, combinatorics, distinct capabilities, classical research, approximation thought, and mathematical physics.