#### 7.13 binomial (21.6.97)

##### 7.13.1 Metha Kamminga

One of my students tried to solve a very simple equation with Maple(release 4).

binomial(8,2)=binomial(x,6)



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?

> binomial(8,2);
28
> binomial(8,6);
28
> GAMMA(9)/GAMMA(7)/GAMMA(3);
28
> binomial(-3,6);
28
> GAMMA(-2)/GAMMA(7)/GAMMA(-8);
Error, (in GAMMA) singularity encountered
> eq:=binomial(8,2)=binomial(x,6);
eq := 28 = binomial(x, 6)
> solve(eq,x);
RootOf(28 - binomial(_Z, 6))
> allvalues(%);
-3.000000000
> fsolve(eq,x);
-3.000000000
> fsolve(eq,x,1..10);
8.000000000
> plot(binomial(x,6),x=-10..10);



##### 7.13.2 Petr Lisonek (23.6.97)

By the very deﬁnition of binomial coeﬃcients, this is a polynomial equation and you must bring it to that form ﬁrst.

> eq:=binomial(8,2)=binomial(x,6);

eq := 28 = binomial(x, 6)

> eq:=expand(eq);

eq := 28 = 1/720 (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x

> solve(eq,x);

1/2 1/2                           1/2 1/2
8, -3, 5/2 + 1/2 (- 43 + 4 I 551   )   , 5/2 - 1/2 (- 43 + 4 I 551   )   ,

1/2 1/2                           1/2 1/2
5/2 + 1/2 (- 43 - 4 I 551   )   , 5/2 - 1/2 (- 43 - 4 I 551   )



##### 7.13.3 Jan-Moritz Franosch (23.6.97)

To ﬁnd out the deﬁnition of the binomial-function in Maple type this

> interface(verboseproc=2);
> print(binomial);



and you will see that

binomial(n,k):=(-1)**k*binomial(k-n-1,k) for n<0, k>0



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:

> fsolve( binomial(x,6)=28, x=0..infinity );
8.000000



##### 7.13.4 Robert Israel (23.6.97)

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.