Binomial coefficient

Binomial coefficient is holomorphic function \(\mathrm{Binomial}\) of two variables defined through function Factorial with

\(\displaystyle \mathrm{Binomial}(x,y)= \frac{x!}{y!\,(x\!-\!y)!}= \frac{\mathrm{Factorial}(x)}{\mathrm{Factorial}(y)\, \mathrm{Factorial}(x\!-\!y)} \)

Function Binomial is incorporated in some programming languages, including Mathematica.

In applications, function Binomial is often used for positive integer values of the first argument and non–negative values of the second argument.

Specific values

\(\mathrm{Binomial}(m,0)=1\)

\(\mathrm{Binomial}(m,1)=m\)

\(\mathrm{Binomial}(m,m\!-\!1)=m\)

\(\mathrm{Binomial}(m,m)=1\)

Pascal triangle

Properties of Factorial lead to the following recurrent relation:

\(\mathrm{Binomial}(m,k)= \mathrm{Binomial}(m-1,k-1)+\mathrm{Binomial}(m-1,k) \)

This property can be used for evaluation of Binomial for integer values of the arguments. Example of the C++ algorithm that does this is shown below:

#include<math.h>
#include<stdio.h>
#define DO(x,y) for(x=0;x<y;x++)
#define N 10
int BC[N+1][N+1];
int main(){ int m,n,k;
 DO(n,N+1) { BC[n][0]=1; BC[n][1]=n; BC[n][n]=1;}
 for(m=2;m<N+1;m++)
 for(n=2;n<N;n++) BC[m][n]=BC[m-1][n-1]+BC[m-1][n];
 DO(m,N+1){ DO(n,N+1) printf("%6d ",BC[m][n]);
           printf("\n");}
}

The resulting output is copypasted below:

     1 0 0 0 0 0 0 0 0 0 0
     1 1 0 0 0 0 0 0 0 0 0
     1 2 1 0 0 0 0 0 0 0 0
     1 3 3 1 0 0 0 0 0 0 0
     1 4 6 4 1 0 0 0 0 0 0
     1 5 10 10 5 1 0 0 0 0 0
     1 6 15 20 15 6 1 0 0 0 0
     1 7 21 35 35 21 7 1 0 0 0
     1 8 28 56 70 56 28 8 1 0 0
     1 9 36 84 126 126 84 36 9 1 0
     1 10 45 120 210 252 210 120 45 10 1

Each number (except the first) in this table appear as sum of the number just above it and that above the previous number. For this reason, such a table is called "Pascal triangle"; some Pascal is believed to be first to employ this property. .

Application

Binomial coefficients appear at the expansion of the integer power of a binom:

\(\displaystyle (a\!+\!b)^n = \sum_{k=0}^n \mathrm{Binomial}(n,k)\, a^k\, b^{n-k} \)

Often, this formula is used for integer \(n\) and commuting objects \(a\) and \(b\).

In principle, the similar formula can be used for non–integer \(n\), but then, the infinite series appears instead of finite sum, and the convergence becomes responsibility of the researcher, who uses it.

References

http://functions.wolfram.com/GammaBetaErf/Binomial/

https://en.wikipedia.org/wiki/Binomial_coefficient

Keywords

Factorial, Hydrogen atom, Hydrogen wave function, Mathematica, Quantum mechanics