Matrices with convolutions of binomial functions and Krawtchouk matrices

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