How to write the product identity in Mathematica?

Questions
1

Is there anybody can help me to write and excute the following product identity in Mathematica?

productidentity#.jpg
productidentity#.jpg414×105 11.7 KB

Thanks a lot for your support

2

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 lot

Regards,
Yahia

3

Hi Yahia,

what you have to do here on the right hand side is - given in utterly <outmost> 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 = 1

1 - x1 = 1 - x1

and for K = 2

(1 - x1) (1 - x2) == 1 - x1 - x2 + x1 x2

x1 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 of

x1 x2 = 1/2 (x1 x2 + x2 x1)

and in general you have an expression as simple as

In[14]:= With[{K = 19}, 
  Product[1 - x[o], {o, 1, K}] == 
   Sum[(-1)^l SymmetricPolynomial[l, Array[x, K]], {l, 0, K}]
  ] // FullSimplify

Out[14]= True

Instead of Subscript[x,1] one used x[1] with advantage, as you see.

Regards
Udo.

4

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

5

To put conditions on terms in Sums and Products you have many possibilities, the simplest is

In[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.

Regards
Udo.

6

Thanks so much Udo for your support..I really appreciate your answer.

regards,
Yahia

7

Hi Yahia,

if you are inclined to implement the formula as it stands, then you might type

Remove[args, cond, X, prod, iter, Y, alghoraniLHS, alghoraniRHS]
args[l_Integer] := 
 ToExpression[StringJoin["o", ToString[#]] & /@ Range[l]]
prod[l_Integer] := Times @@ Thread[x[args[l]]]
cond[l_Integer] := 
 If[And @@ (X @@@ Union[Sort /@ Permutations[args[l], {2}]]), 1, 0]
iter [m_Integer, l_Integer] := {#, 1, m} & /@ args[l] /; m >= l
alghoraniLHS[K_Integer?Positive] := Product[(1 - x[o]), {o, 1, K}]
alghoraniRHS[K_Integer?Positive] := 
 1 + Sum[(-1)^l/l! Y[cond[l] prod[l], Sequence @@ iter[K, l]], {l, 1, 
     K}] /. {Y :> Sum, X :> Unequal}

into your keyboard. The args function ensures that all functions have the same variables in use. See an example

In[94]:= alghoraniRHS[4]

Out[94]= 1 - x[1] - x[2] - x[3] - x[4] + x[1] x[2] x[3] x[4] + 
 1/2 (2 x[1] x[2] + 2 x[1] x[3] + 2 x[2] x[3] + 2 x[1] x[4] + 
    2 x[2] x[4] + 2 x[3] x[4]) + 
 1/6 (-6 x[1] x[2] x[3] - 6 x[1] x[2] x[4] - 6 x[1] x[3] x[4] - 
    6 x[2] x[3] x[4])

Check a case - it runs considerably longer than the SymmetricPolynomial[], of course - all the permutations have to be sorted out

In[100]:= Expand[alghoraniLHS[7]] - alghoraniRHS[7] // Simplify

Out[100]= 0

The multiple sums to find terms one knows in advance are computing time wasters

In[109]:= Expand[alghoraniLHS[9]] - alghoraniRHS[9] // Simplify

Out[109]= 0

Please note that even alghoraniRHS[] doesn’t go the whole nine yards of implementing the formula because it has the l = 0 term, the 1, in front of it. Can you find a solution without that?

Pleaso note also, that Union[Sort /@ Permutations[…]] is bad style, because it filters out … permutations.

Regards
Udo.

8

You can also take advantage of the relatively new iterator conventions:

Product[1 - x[l], {l, 1, k}] ==  Sum[(-1)^l*Product[x[t], {t, n}], {l, 0, k}, {n, Subsets[Range[k], {l}]}]

This is another way to implement index restrictions, with programming constructs, e.g.,

Product[1 - x[l], {l, 1, k}] ==
Sum[(-1)^l/l! Product[x[t], {t, n}], {l, 0, k}, {n, Select[Tuples[Range[k], l], UnsameQ @@ # &]}]
9

Thanks!