Multinomial convolution polynomials

AbstractIn [9] Knuth shows how to derive the convolution formulas of Hagen, Rothe and Abel from Vandermonde's convolution and the binomial theorem for integer exponents. In the present paper, we shall first present a short and elementary proof of the multi-extension of the above convolution formulas, due to Raney and Mohanty. In the second part we shall present a multi-version of Knuth's approach to convolution polynomials and derive another short proof of the above formulas