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.