There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the “binomial series” rather than “binomial theorem.”
The most general case of the binomial theorem is the binomial series identity
(x+a)^nu=sum_(k=0)^infty(nu; k)x^ka^(nu-k),
(1)
where (nu; k) is a binomial coefficient and nu is a real number. This series converges for nu>=0 an integer, or |x/a|<1. This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with a=1 the binomial theorem.
When nu is a positive integer n, the series terminates at n=nu and can be written in the form
(x+a)^n=sum_(k=0)^n(n; k)x^ka^(n-k).
(2)
This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).
The differing terminologies are summarized in the following table.
“binomial theorem” source
(x+a)^nu=sum_(k=0)^(infty)(nu; k)x^ka^(nu-k) Graham et al. (1994, p. 162)
(x+1)^nu=sum_(k=0)^(infty)(nu; k)x^k Arfken (1985, p. 307)
(x+a)^n=sum_(k=0)^(n)(n; k)x^ka^(n-k) Abramowitz and Stegun (1972, p. 10)
The binomial theorem was known for the case n=2 by Euclid around 300 BC, and stated in its modern form by Pascal in a posthumous pamphlet published in 1665. Pascal’s pamphlet, together with his correspondence on the subject with Fermat beginning in 1654 (and published in 1679) is the basis for naming the arithmetical triangle in his honor.
Newton (1676) showed the formula also holds for negative integers -n,
(x+a)^(-n)=sum_(k=0)^infty(-n; k)x^ka^(-n-k),
(3)
which is the so-called negative binomial series and converges for |x|<a.
In fact, the generalization
(1+z)^a=sum_(k=0)^infty(a; k)z^k
(4)
holds for all complex z with |z|<1.
Among his many other talents, Major General Stanley in Gilbert and Sullivan’s operetta the Pirates of Penzance impresses the pirates with his knowledge of the binomial theorem in “The Major General’s Song” as follows: “I am the very model of a modern Major-General, I’ve information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I’m very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I’m teeming with a lot o’ news-- With many cheerful facts about the square of the hypotenuse.”
