Mma 4.0 is known (by Bill G.) to daringly cut corners, but gives: In[1]:=Sum[o! p! q! r!/(o+p+q+r)!,{r,0,\[Infinity]}] Out[1]=(o!*p!*q!)/((-1 + o + p + q)^2*Gamma[-1 + o + p + q]) and that confirms your o!p!q!/(o+p+q-1)(o+p+q-1)! W. ----- Original Message ----- From: "Fred lunnon" <fred.lunnon@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Sunday, August 29, 2010 7:15 PM Subject: Re: [math-fun] sums of reciprocal multinomials
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!
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