The problem of representing an integer as a sum of squares of
integers is one of the oldest and most significant in mathematics.
It goes back at least 2000 years to Diophantus, and continues more
recently with the works of Fermat, Euler, Lagrange, Jacobi,
Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's
elliptic function approach dates from his epic Fundamenta Nova of
1829. Here, the author employs his combinatorial/elliptic function
methods to derive many infinite families of explicit exact formulas
involving either squares or triangular numbers, two of which
generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or
4n(n+1) squares, respectively, without using cusp forms such as
those of Glaisher or Ramanujan for 16 and 24 squares. These results
depend upon new expansions for powers of various products of
classical theta functions. This is the first time that infinite
families of non-trivial exact explicit formulas for sums of squares
have been found.
The author derives his formulas by utilizing combinatorics to
combine a variety of methods and observations from the theory of
Jacobi elliptic functions, continued fractions, Hankel or Turanian
determinants, Lie algebras, Schur functions, and multiple basic
hypergeometric series related to the classical groups. His results
(in Theorem 5.19) generalize to separate infinite families each of
the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert
series expansions of classical theta functions in sections 40-42 of
the Fundamental Nova. The author also uses a special case of his
methods to give a derivation proof of the two Kac and Wakimoto
(1994) conjectured identities concerning representations of
apositive integer by sums of 4n2 or 4n(n+1) triangular numbers,
respectively. These conjectures arose in the study of Lie algebras
and have also recently been proved by Zagier using modular forms.
George Andrews says in a preface of this book, This impressive work
will undoubtedly spur others both in elliptic functions and in
modular forms to build on these wonderful discoveries'.
Audience: This research monograph on sums of squares is
distinguished by its diversity of methods and extensive
bibliography. It contains both detailed proofs and numerous
explicit examples of the theory. This readable work will appeal to
both students and researchers in number theory, combinatorics,
special functions, classical analysis, approximation theory, and
mathematical physics.
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