It turns out that what my physicist friend really wanted was the following k-fold nested summation, where each upper limit is infinite and each lower limit the next outer summation variable: Given integers n > 0, k > 0, show \sum_{i_k = 0}^oo (1/i_k) ... ... \sum_{i_2 = i_3}^oo (1/i_2) \sum_{i_1 = i_2}^oo (1/i_1) (n+1)! (i_1)! / (n+1+i_1)! = (n+1)/n^k . I know of no way even to present the left-hand side to a CAS, let alone persuade the beast to deliver the correct answer: the employment of a recursive function appears unavoidable, then there are tricky issues of delayed evaluation involved. Anyone care to comment? Fred Lunnon On Thu, Aug 12, 2010 at 6:11 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
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!