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# How to write the product identity in Mathematica?

Posted 11 years ago
 Is there anybody can help me to write and excute the following product identity in Mathematica?Thanks a lot for your support
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Posted 11 years ago
 To put conditions on terms in Sums and Products you have many possibilities, the simplest isIn[18]:= Sum[If[o1 != o2 && o1 != o3 && o1 != o4              && o2 != o3 && o2 != o4              && o3 != o4, 1, 0] x[o1] x[o2] x[o3] x[o4],              {o1, 1, 2}, {o2, 1, 3}, {o3, 1, 4}, {o4, 1, 5}]Out[18]= 8 x[1] x[2] x[3] x[4] + 4 x[1] x[2] x[3] x[5] + 2 x[1] x[2] x[4] x[5] + x[1] x[3] x[4] x[5] + x[2] x[3] x[4] x[5]You must not necessarily write such an ulgy If[], there is the KroneckerDelta[], there are totally antisymmetric tensors built-in for some of the standard cases in filtered sums and/or products.RegardsUdo.
Posted 11 years ago
 Hi Yahia,what you have to do here on the right hand side is - given in utterly ugly notation in the exercise - simply to sum up the symmetric functions of {x1, x2, ..., xl}of 1, 2, ..., l arguments augmented with the right sign. So you have for K = 11 - x1 = 1 - x1and for K = 2(1 - x1) (1 - x2) == 1 - x1 - x2 + x1 x2x1 and x2 are the both symmetric functions of 1 argument, x1 x2 is the single symmteric function of two arguments in {x1,x2}. Please note that using the SymmetricPolynomial allows you to get rid of the Factorial[] because ofx1 x2 = 1/2 (x1 x2 + x2 x1)and in general you have an expression as simple asIn[14]:= With[{K = 19},   Product[1 - x[o], {o, 1, K}] ==    Sum[(-1)^l SymmetricPolynomial[l, Array[x, K]], {l, 0, K}]  ] // FullSimplifyOut[14]= TrueInstead of Subscript[x,1] one used x[1] with advantage, as you see.RegardsUdo.
Posted 11 years ago
 However, I have assume that K=2, and have got a result as follows In[82]:= Sum[(-1)^l/l!* Sum[Product[Subscript[x, Subscript[n, t]], {t, 1, l}] Boole[ Subscript[n, 1] != Subscript[n, 2]], {Subscript[n, 1], 1, 2}, {Subscript[n, 2], 1, 2}], {l, 0, 2}]                         Out[82]= 2 - Subscript[x, 1] - Subscript[x, 2] + Subscript[x, 1] Subscript[x, 2]Which is not match with the fact that In[83]:= Product[1 - Subscript[x, l], {l, 1, 2}]Out[83]=1 - Subscript[x, 1] - Subscript[x, 2] + Subscript[x, 1] Subscript[x, 2]I think there is something wrong with the method of putting exclusions such as Boole on the above Sum. I will be happy to get your suggestions.Thanks a lotRegards,Yahia
Posted 11 years ago
 Thanks for your interest. In fact, the target of my question how to put condition or exclusions to Sum or product when you have such equations?..regards,Yahia
Posted 11 years ago