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Mathematics Mathematica Wolfram Language Symbolic Computations

Get right results with binomial functions in MMA 7?

Hans Dolhaine

Hans Dolhaine, retired

Posted 9 years ago

Hello all, I am using Mathematica version 7 and would like to ask a question to those using newer versions. The expression t1 = Sum[ j Binomial[A, j] Binomial[B, n - j], {j, 0, n}] can be calculated by pencil and paper giving as result A Binomial[ A + B - 1, n - 1] while evaluating the expression for t1 (on my system) yields -(((-1)^n A (-1 - A - B + n)!)/((-A - B)! (-1 + n)!)) Feeding numbers to it gives In[58]:= A Binomial[ A + B - 1, n - 1] /. A -> 7 /. B -> 5 /. n -> 4 Out[58]= 1155 but In[60]:= t1 /. A -> 7 /. B -> 5 /. n -> 4 Out[60]= 0 Obviously something is severely wrong! The same thing comes along with In[15]:= t2 = Sum[ j^2 Binomial[A, j] Binomial[B, n - j], {j, 0, n}] Out[15]= ((-1)^n A (B - n + A n) (-1 - A - B + n)!)/((1 - A - B)! (-1 + n)!) which is evaluated by the paper and pencil method to A (A - 1 + (A + B - 1)/(n - 1)) Binomial[A + B - 2, n - 2] with striking differences when you feed numbers in. What do the newer versions of Mma say to this? Regards, Hans Dolhaine

POSTED BY: Hans Dolhaine

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Valeriu Ungureanu

Valeriu Ungureanu, Moldova State University

Posted 9 years ago

MMa 11 says: In[1]:= t1 = Sum[j Binomial[A, j] Binomial[B, n - j], {j, 0, n}] Out[1]= A Binomial[-1 + A + B, -1 + n] In[2]:= % /. {A -> 7, B -> 5, n -> 4} Out[2]= 1155 In[3]:= t2 = Sum[j^2 Binomial[A, j] Binomial[B, n - j], {j, 0, n}] Out[3]= ((-1)^n A (B - n + A n) (-1 - A - B + n)!)/((1 - A - B)! (-1 + n)!)

MMa 11 says:

In[1]:= t1 = Sum[j Binomial[A, j] Binomial[B, n - j], {j, 0, n}]

Out[1]= A Binomial[-1 + A + B, -1 + n]

In[2]:= % /. {A -> 7, B -> 5, n -> 4}

Out[2]= 1155

In[3]:= t2 = Sum[j^2 Binomial[A, j] Binomial[B, n - j], {j, 0, n}]

Out[3]= ((-1)^n A (B - n + 
   A n) (-1 - A - B + n)!)/((1 - A - B)! (-1 + n)!)

POSTED BY: Valeriu Ungureanu

Hans Dolhaine

Hans Dolhaine, retired

Posted 9 years ago

Hello, thanks. That is sort of much better, but the result for t2 is still, say, less than optimal. (Try to give numbers into t2 and compare this with the correct pencil&paper result)