Matrices with convolutions of binomial functions and Krawtchouk matrices

We introduce a class S_N of matrices whose elements are terms of convolutions of binomial functions of complex numbers. A multiplication theorem is proved for elements of S_N. The multiplication theorem establishes a homomorphism of the group of 2 by 2 nonsingular matrices with complex elements into a group G_N contained in S_N. As a direct consequence of representation theory, we also present related spectral representations for special members of G_N. We show that a subset of G_N constitutes the system of Krawtchouk matrices, which extends published results for the symmetric case.