Some binomial convolution formulas (Q2785024)

Let \(d\), \(m\), \(n\) be positive integers. Let \(\{A(k)\}_k\), \(\{B(k)\}_k\) be two sequences of binomial coefficients. When \(d= 2,3\) and \(4\), the convolutions of Vandermonde type \(\sum_k A(m+ dk)B(n- dk)\) are called duplicate, triplicate and quadruplicate, respectively. In 1999, using the formal power series method, the author gave some binomial formulas on duplicate convolutions. In the present paper, some binomial formulas on triplicate and quadruplicate convolutions are given.