Comment by Christian Krattenthaler, May 12, 2008:
In the same way, Formula (3) is equivalent to the second main result in "Some convolutions with multinomial coefficients and related probability distributions" by S. G. Mohanty (SIAM Review 8 (1966), 501-509).
These convolution type formulas are given coverage also in Mohanty's book "Lattice Path Counting and Applications" (Academic Press, 1979) in Chapter 6, Section 2. It is also briefly hinted at how these convolution formulas follow rather easily by decomposition of paths (see Chapter 4, page 89).
Editor's note: The paper was accepted Mar 25, 2008 not Mar 25, 2007 as stated in the paper.