One of my students tried to solve a very simple equation with Maple(release 4).
Maple gave the solution \(x=-3\) but by heart you can see it must be \(x=6\).
Can anybody explain me the structure of the binomialfunction in case of negative integers?
By the very deﬁnition of binomial coeﬃcients, this is a polynomial equation and you must bring it to that form ﬁrst.
To ﬁnd out the deﬁnition of the binomial-function in Maple type this
and you will see that
This is simply the formular
binomial(n,k)=n*(n-1)*...*(n-k+1) / k*(k-1)*...*1 for n<0.
The help-function ?binomial does not tell the whole truth in this case.
A workaround would be to let fsolve only search for positive solutions:
The general deﬁnition (which works for any complex n, and nonnegative integer m) is binomial(n,m) = n (n-1) ... (n-m+1)/m!
Thus binomial(-n,m) = (-1)^m binomial(n+m,m).
You might look at Graham, Knuth and Patashnik, ”Concrete Mathematics”, for more discussion and applications of this.