A new Wannier function made from distributed Gaussians is reported. Convolution integrals of the Wannier functions are studied and analytical expressions for the convolution integral are given. A new expression for a set of coefficients introduced in an earlier publication [J. Math. Phys. 38, 4815 (1997)] is derived. Using this new expression, the peculiar behavior of another set of coefficients introduced in the same publication in a particular limit is explained. (C) 2005 American Institute of Physics.