Binomial Transforms and the Binomial Convolution of Sequences

Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving Fibonacci numbers, Bernoulli numbers, Catalan numbers, harmonic numbers, odd harmonic numbers, Stirling numbers of the second kind, and binomial coefficients. In addition, we present several results which allow the construction of new binomial-transform pairs from existing ones. Many new relations concerning self-inverse sequences are also derived.