## Infinite Families of Exact Sums of Squares Formulas, Jacobi by Stephen C. Milne

By Stephen C. Milne

The challenge of representing an integer as a sum of squares of integers is likely one of the oldest and most vital in arithmetic. It is going again at the very 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 technique dates from his epic **Fundamenta Nova** of 1829. right here, the writer employs his combinatorial/elliptic functionality the way to derive many endless households of specific certain 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 kinds equivalent 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 capabilities. this is often the 1st time that limitless households of non-trivial distinct specific formulation for sums of squares were came across.

The writer derives his formulation by using combinatorics to mix numerous tools and observations from the idea of Jacobi elliptic features, persevered fractions, Hankel or Turanian determinants, Lie algebras, Schur features, and a number of uncomplicated hypergeometric sequence regarding 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 different case of his ways to supply a derivation evidence of the 2 Kac and Wakimoto (1994) conjectured identities bearing on representations of a good 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 kinds. George Andrews says in a preface of this publication, `This striking paintings will surely spur others either in elliptic capabilities and in modular kinds to construct on those fabulous discoveries.'

*Audience:* This examine monograph on sums of squares is amazing via its range of equipment and large bibliography. It comprises either exact proofs and diverse particular examples of the idea. This readable paintings will attract either scholars and researchers in quantity idea, combinatorics, specific services, classical research, approximation idea, and mathematical physics.