Here's something which may be as new to some other people on the list as it was to me, though it must surely be well-known (and probably has some connection to do with those ghastly hypergeometric series that I keep trying to avoid). I illustrate with the tetranomial case --- the generalisation to any dimension is obvious: \sum_{0<=j<oo} p!q!r!j!/(p+q+r+j)! = p!q!r!/(p+q+r-1)(p+q+r-1)! Does anyone know of a reference for these and similar results --- for example, generalisation to summation over several indices? Can Mathematica, Macsyma etc. deduce them --- Maple 9 flunked! This was brought to my attention by an enquiry from a physicist: not only did I have to admit that I didn't recognise it, but neither could I tell him where to go look for it. Fred Lunnon