Kronecker-type identities and formulas for sums of squares and sums of triangular numbers

We recall Kronecker's identity and review how limiting cases give the representations of a number as
a sum of four squares and the representations of a number as a sum of two squares.  The two formulas
imply respectively Lagrange's theorem that every number can be written as a sum of four squares and
Fermat's theorem that an odd prime can be written as the sum of two squares if and only if it is congruent
to 1 modulo 4.  By considering a limiting case of a higher-dimensional Kronecker-type identity, we obtain
an identity found by both Andrews and Crandall.  We then use the Andrews-Crandall identity to give a
new proof of a formula of Gauss for the representations of a number as a sum of three squares.  From
the Kronecker-type identity, we also deduce Gauss's theorem that every positive integer is representable
as a sum of three triangular numbers.