  In this paper we derive many infinite families of explicit exact formulas
involving either squares or triangular numbers, two of which generalize
Jacobi's 4 and 8 squares identities to 4n^2 or 4n(n+1) squares, respectively,
without using cusp forms. In fact, we similarly generalize to infinite families
all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions
of classical theta functions. In addition, we extend Jacobi's special analysis
of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles,
20 squares, 20 triangles, respectively. These results, depending on new
expansions for powers of various products of classical theta functions, arise
in the setting of Jacobi elliptic functions, associated continued fractions,
regular C-fractions, Hankel or Turanian determinants, Fourier series, Lambert
series, inclusion/exclusion, Laplace expansion formula for determinants, and
Schur functions. The Schur function form of these infinite families of
identities are analogous to the eta-function identities of Macdonald. Moreover,
the powers 4n(n+1), 2n^2+n, 2n^2-n that appear in Macdonald's work also arise
at appropriate places in our analysis. A special case of our general methods
yields a proof of the two Kac--Wakimoto conjectured identities involving
representing a positive integer by sums of 4n^2 or 4n(n+1) triangular numbers,
respectively.
