Binomial theorem ( $(x + y)^{n} = \sum_{k = 0}^{n} (k n) x^{k} y^{n - k}$ )

, the number of subsets of a five-element set. Each member has two inclusion choices, agreeing with the rule of product and the cardinality of the power set.

Table of contents

Binomial expansion

 0  0

The binomial expansion expresses

(a + b)^{n}

\sum_{k = 0}^{n} (k n) a^{n - k} b^{k}

for a nonnegative integer

n

. The binomial coefficients give the number of ways to choose which

k

factors supply

b

Binomial inversion ( $b_{n} = \sum_{k = 0}^{n} (- 1)^{n - k} (k n) a_{k} ⟺ a_{n} = \sum_{k = 0}^{n} (k n) b_{k}$ )

 0  0

These two finite-sum transformations are inverses over any commutative ring. Substitute one sum into the other and use

(k n) (j k) = (j n) (k - j n - j)

. The inner alternating sum is

(1 - 1)^{n - j}

by the binomial theorem, so only

j = n

survives. In a Mahler expansion,

b_{n}

is the nth finite difference of the sampled values

a_{k}

Alternating binomial moment ( $S_{r} (n) = \sum_{m = 0}^{n} (- 1)^{m} m^{r} (m n)$ )

 0  0

For nonnegative integers

r, n

, the alternating binomial moment satisfies

S_{r} (n) = 0

when

r < n

and

S_{n} (n) = (- 1)^{n} n!

. Expand the polynomial

X^{r}

in falling factorials and differentiate the binomial theorem to prove these identities. Equivalently, alternating binomial coefficients implement a finite difference that annihilates polynomials of degree less than

n

Freshman's dream

 1  0

In a commutative ring of characteristic

p

(x + y)^{p} = x^{p} + y^{p} .

(1)

Iteration gives

(x + y)^{p^{n}} = x^{p^{n}} + y^{p^{n}}

Binomial theorem for commuting matrices

 0  0

If square matrices

A

and

B

commute, then

(A + B)^{n} = \sum_{k = 0}^{n} (k n) A^{n - k} B^{k} .

(1)

The usual proof works because commutativity allows words with the same numbers of

A

and

B

factors to be collected.

Alternating binomial-square sum

 0  0

Coefficient extraction from

(1 - x)^{n} (1 + x)^{n} = (1 - x^{2})^{n}

gives

\sum_{r = 0}^{n} (- 1)^{r} (r n)^{2} = {0, (- 1)^{n /2} (n /2 n), n odd, n even .

(1)

 Synonyms (1)

codex/binomial-formula

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Binomial theorem by

Wikipedia Bot  1

 View more

The Binomial Theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this formula: - $\sum$ denotes summation.

 Read the full article

  See all articles in the same topic Create my own version

Binomial theorem ( $(x + y)^{n} = \sum_{k = 0}^{n} (k n) x^{k} y^{n - k}$ )

Binomial expansion

Binomial inversion ( $b_{n} = \sum_{k = 0}^{n} (- 1)^{n - k} (k n) a_{k} ⟺ a_{n} = \sum_{k = 0}^{n} (k n) b_{k}$ )

Alternating binomial moment ( $S_{r} (n) = \sum_{m = 0}^{n} (- 1)^{m} m^{r} (m n)$ )

Freshman's dream

Binomial theorem for commuting matrices

Alternating binomial-square sum

 Ancestors (5)

 Incoming links (145)

 Synonyms (1)

 Discussion (0) New discussion

 Articles by others on the same topic (1)

 Discussion (0)  Subscribe (1) New discussion

 Discussion (0) New discussion