Binomial coefficients

The coefficients at the powers of $ z $ in the decomposition of the Newton binomial $ {(1 + z) } ^ {N} $. The binomial coefficients are denoted by

$$ \left ( \begin{array}{c} N \\ n \end{array} \right ) $$

or by $ C _ {N} ^ { n } $, and are given by

$$ \tag{1 } \left ( \begin{array}{c} N \\ n \end{array} \ \right ) = C _ {N} ^ { n } = \ \frac{N! }{n! (N-n)! } = $$

$$ = \ \frac{N (N - 1) \dots (N - n + 1) }{n! } ,\ 0 \leq n \leq N. $$

The first-mentioned notation is due to L. Euler; the notation $ C _ {N} ^ { n } $ appeared in the 19th century, and is probably connected with the interpretation of the binomial coefficients $ C _ {N} ^ {n } $ as the number of distinguishable non-ordered combinations (cf. Combination) from $ N $ different objects with $ n $ objects in each combination. Binomial coefficients are most conveniently written as numbers in the arithmetical triangle, or Pascal triangle, the construction of which is based on the following property of binomial coefficients:

$$ \tag{2 } \left ( \begin{array}{c} N \\ n \end{array} \right ) + \left ( \begin{array}{c} N \\ n+1 \end{array} \right ) = \ \left ( \begin{array}{c} N+1 \\ n+1 \end{array} \right ) . $$

Binomial coefficients, as well as the arithmetical triangle, were known concepts to the mathematicians of antiquity, in more or less developed forms. B. Pascal (l665) conducted a detailed study of binomial coefficients. The binomial coefficients are also connected by many useful relationships other than (2), for example:

$$ \tag{3a } \left ( \begin{array}{c} N \\ n \end{array} \right ) = \left ( \begin{array}{c} N \\ N-n \end{array} \right ) ; $$

$$ \tag{3b } \left ( \begin{array}{c} N \\ n \end{array} \right ) = \sum _ {k = 0 } ^ { n } \left ( \begin{array}{c} m \\ k \end{array} \right ) \left ( \begin{array}{c} N-m \\ n-k \end{array} \right ) ,\ \ n \leq m \leq N - n; $$

$$ \tag{3c } \sum _ {k = 0 } ^ { N } (-1) ^ {k} k ^ {m} \left ( \begin{array}{c} N \\ k \end{array} \right ) = 0,\ \ m = 0 \dots N - 1 ; $$

$$ \tag{3d } \sum _ {k = 0 } ^ { N } \left ( \begin{array}{c} n \\ k \end{array} \right ) ^ {2} = \left ( \begin{array}{c} 2n \\ n \end{array} \right ) ; $$

$$ \tag{3e } \sum _ {k = 0 } ^ { N } \left ( \begin{array}{c} N \\ k \end{array} \right ) k (k - 1) \dots (k - r + 1) = $$

$$ = \ N (N - 1) \dots (N - r + 1) \cdot 2 ^ {N - r } ; $$

$$ \tag{3f } \sum _ {k = 0 } ^ { N } (-1) ^ {k} \left ( \begin{array}{c} N \\ k \end{array} \right ) k (k-1) \dots (k - r + 1) = 0 . $$

In particular, (3a)–(3f) yields

$$ \tag{4 } \left . \begin{array}{l} \sum _ {k = 0 } ^ { N } \left ( \begin{array}{c} N \\ k \end{array} \right ) = 2 ^ {N} ; \\ \sum _ {k = 0 } ^ { N } (-1) ^ {k} \left ( \begin{array}{c} N \\ k \end{array} \right ) = 0. \\ \end{array} \ \right \} $$

The use of the Stirling formula yields approximate expressions for binomial coefficients. Thus, if $ N $ is much larger than $ n $:

$$ \left ( \begin{array}{c} N \\ n \end{array} \right ) \sim \frac{N ^ {n} }{n! } . $$

In the case of a complex number $ \alpha $, binomial coefficients are generalized according to the formula

$$ \left ( \begin{array}{c} \alpha \\ n \end{array} \ \right ) = \ \frac{\alpha ( \alpha - 1) \dots ( \alpha - n + 1) }{n! } ,\ \ n > 0; \ \ \left ( \begin{array}{c} \alpha \\ 0 \end{array} \ \right ) = 1. $$

In this generalization, some of the relations (2)–(4) are preserved, but usually in a modified form. For instance,

$$ \left ( \begin{array}{c} \alpha \\ n \end{array} \ \right ) + \left ( \begin{array}{c} \alpha \\ n + 1 \end{array} \ \right ) = \left ( \begin{array}{c} {\alpha + 1 } \\ {n + 1 } \end{array} \ \right ) ; $$

$$ \sum _ {k = 0 } ^ { {+ } \infty } \left ( \begin{array}{c} \alpha \\ k \end{array} \right ) = 2 ^ \alpha ,\ \mathop{\rm Re} \alpha > -1; $$

$$ \sum _ {k = 0 } ^ { {+ } \infty } (-1) ^ {k} \left ( \begin{array}{c} \alpha \\ k \end{array} \right ) = 0,\ \mathop{\rm Re} \alpha > 0. $$

For tables of binomial coefficients see [2], [3].

References

[1]	G.A. Korn, T.M. Korn, "Mathematical handbook for scientists and engineers" , McGraw-Hill (1968) Zbl 0177.29301
[2]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) Zbl 0529.62099
[3]	J.C.P. Miller (ed.) "Table of binomial coefficients". Royal Society Mathematical Tables 3 Cambridge University Press (1954) Zbl 0059.11301