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whenever n is any non-negative integer and the numbers

are the binomial coefficients. This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was however known long before to Chinese mathematician Yang Hui.

For example, here are the cases n=2, n=3 and n=4:

(x + y)² = x² + 2xy + y²

(x + y)³ = x³ + 3x²y + 3xy² + y³

(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a ring as long as xy = yx.

1 Newton's generalized binomial theorem

where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by

(which in case k = 0 is a product of no numbers at all and therefore equal to 1, and in case k = 1 is equal to r, as the additional factors (r − 1), etc., do not appear in that case).

For a more extensive account of Newton's generalized binomial theorem, see binomial series.

The sum in (2) converges and the equality is true whenever the real or complex numbers x and y are "close together" in the sense that the absolute valueIn mathematics, the absolute value (or modulus of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3. Definition It can be defined as follows: For any real number a the absolute value of a deno | x/y | is less than one.

The geometric series is a special case of (2) where we choose y = 1 and r = −1.

Formula (2) is also valid for elements x and y of a Banach algebraIn functional analysis, a Banach algebra named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related as long as xy = yx, y is invertible and ||x/y|| < 1.

2 "Binomial type"

The binomial theorem can be stated by saying that the polynomial sequenceIn mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3,. in which each index is equal to the degree of the corresponding polynomial. Various special polynomial sequences are known by eponyms; amon

is of binomial typeDefinition In mathematics, a polynomial sequence, i. a sequence of polynomials indexed by { 0, 1, 2, 3,. in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities : Many such sequenc.