Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x(2) + ey(2) + fz(2) + gt(2) with e, f, g is an element of {1, 2, 4, 8}.

Some new theta function identities are proved and used to determine the number of representations of a positive integer n by certain quaternary quadratic forms.

The convolution sums Sigma(m < n/5) sigma(m)sigma(n - 5m) and Sigma(m </7) sigma(m)sigma(n - 7m) are evaluated for all n is an element of N.