# Summations on the same expression do not give the same result?

Posted 1 year ago
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 Hey guys:When I tried to take summation with respect to my expression Pmoe, I found that the Mathematica does not give consistent results if I put the summation symbol in different places. Please see my attachment where I use Pmoe1 and Pmoe2 to represent two different ways of summation, which should give the same result. It turns out that Pmoe1 is correct if you let Mathematica give values for each i and then sum all the values up. Appreciate anyone who can explain it a bit more.Thanks in advance. Attachments:
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Posted 1 year ago
 In your definition of Pmoe if you change = to := then the results of the two tables are identical.If you study the help pages for = and := then perhaps this will help you begin to understand the difference.The difference in your example depends on when the right hand side of Pmoe will be evaluated.
Posted 1 year ago
 Sorry I am not convinced.My question is why Pmoe1 and Pmoe2 are different, which are both defined by :=.My understanding for := is that the expression is evaluated on request and for = is that the expression is evaluated immediately. However, Pmoe1 and Pmoe2 equal in theoretical sense, which imply they have the values no matter when they are evaluated. I also tried to increase the precision level and the result do not change.
 I have a suspicion something is funky with the symbolic solution to the inner sum that produces the Hypergeometric1F1. When you use Set to define Pmoe, then Simplify[Pmoe1[A, \[Alpha], \[Beta], t, 1] == Pmoe2[A, \[Alpha], \[Beta], t, n] /. n -> 1] is True, but neither equals Pmoe2[A, \[Alpha], \[Beta], t, 1] that has already had the symbolic sum taken. On the other hand, using SetDelayed to define Pmoe we have Simplify[Pmoe1[A, \[Alpha], \[Beta], t, 1] == Pmoe2[A, \[Alpha], \[Beta], t, 1]] is True, but neither equals Pmoe2[A, \[Alpha], \[Beta], t, n] /. n -> 1. I think this means there is agreement only when the inner sum is either taken first or not taken first, but I am honestly a bit confused here.