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The multinomial theorem for finitely many complex variables

Statement

Let $m, n \in N$ , let $W (n, m)$ and $(\binom{n}{α})$ be as in The multinomial coefficient $(\binom{n}{k_{0}, \dots, k_{m - 1}})$ as the number of ordered partitions of an $n$ -set into blocks of prescribed sizes, and let $z_{0}, \dots, z_{m - 1} \in C$ . Write $ι_{C} : N \to C$ for the canonical natural of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field. Then

${(\sum_{i < m} z_{i})}^{n} = \sum_{α \in W (n, m)} ι_{C} ⁣ ((\binom{n}{α})) \prod_{i < m} z_{i}^{α_{i}} .$

For every $α \in W (n, m)$ , the same coefficient satisfies $ι_{C} ⁣ ((\binom{n}{α})) \prod_{i < m} ι_{C} (α_{i}!) = ι_{C} (n!) .$ The sum on the right is the finite sum in the additive commutative monoid of $C$ ; the statement includes $m = 0$ and $n = 0$ .

Facts & Assumptions

[F1]

$W (n, m)$ is finite and $(\binom{n}{α})$ is a natural number. When $m = 0$ , $W (0, 0) = {()}$ and $W (n, 0) = \emptyset$ for $n \geq 1$ (The multinomial coefficient $(\binom{n}{k_{0}, \dots, k_{m - 1}})$ as the number of ordered partitions of an $n$ -set into blocks of prescribed sizes, The multinomial coefficient equals $n! / \prod_{i < m} k_{i}!$ , and $(x_{0} + \dots + x_{m - 1})^{n} = \sum ι ⁣ (\binom{n}{k}) \prod_{i < m} x_{i}^{k_{i}}$ in $R$ ).

[F2]

For $x, y \in C$ and $j \in N$ , $(x + y)^{j} = \sum_{k \leq j} ι_{C} ⁣ ((\binom{j}{k})) x^{k} y^{j - k}$ (The binomial theorem over the complex field).

[F3]

For $α \in W (n, m)$ , $(\binom{n}{α}) \prod_{i < m} α_{i}! = n!$ ; and if $j \leq n$ , then $j + (n - j) = n$ and $(\binom{n}{j}) j! (n - j)! = n!$ (Order on the natural numbers, The multinomial coefficient equals $n! / \prod_{i < m} k_{i}!$ , and $(x_{0} + \dots + x_{m - 1})^{n} = \sum ι ⁣ (\binom{n}{k}) \prod_{i < m} x_{i}^{k_{i}}$ in $R$ , $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ).

[F4]

Every factorial $j!$ is a nonzero natural number (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[F5]

Multiplication in $N$ cancels a common nonzero factor (Cancellation for multiplication by a nonzero factor).

[F6]

The canonical natural $ι_{C}$ is defined by $ι_{C} (0) = 0$ and $ι_{C} (j + 1) = ι_{C} (j) + 1$ ; complex integer powers use $z^{0} = 1$ and $z^{j + 1} = z^{j} z$ (The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Integer powers in the complex field).

[F7]

The additive structure of $C$ is a commutative monoid. Its finite sums over finite sets are independent of enumeration and invariant under bijective reindexing; finite products in the multiplicative monoid use the same finite-list recursion (Semigroup and monoid, $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , A finite sum in a commutative monoid indexed by an arbitrary finite set, Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule, The product $g_{0} g_{1} \dots g_{n - 1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity).

[F8]

The natural operations satisfy $a + 0 = a$ , $a + (j + 1) = (a + j) + 1$ , $a \cdot 0 = 0$ , and $a \cdot (j + 1) = a \cdot j + a$ (Addition of natural numbers, Multiplication of natural numbers).

[F9]

Complex field arithmetic is associative and distributive, and integer powers of nonzero elements obey the usual exponent laws ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Laws of integer exponents).

[F10]

Induction on $N$ is valid (The principle of mathematical induction).

Proof

technique · induction on the number of variables, with the complex binomial theorem in the inductive step

Given: Naturals $m, n$ , the finite index set $W (n, m)$ , and complex numbers $z_{i}$ for $i < m$ .

1.1F3F6F7F8F9F10given

The canonical natural preserves addition and multiplication in $C$ . For fixed $a \in N$ , induction [F10] on $b$ proves $ι_{C} (a + b) = ι_{C} (a) + ι_{C} (b)$ : the base case uses $a + 0 = a$ , and the successor case uses $a + (b + 1) = (a + b) + 1$ and the recursion in [F6]. A second induction on $b$ proves $ι_{C} (a b) = ι_{C} (a) ι_{C} (b)$ : at $b = 0$ both sides are $0$ , and at the successor use $a (b + 1) = a b + a$ , the first identity, and distributivity in [F9]. Also $ι_{C} (1) = 1$ . Applying [F3] and the multiplicative property just proved successively along the finite product recursion [F7] gives the coefficient identity in the statement.

1.2F1F6given

For $m = 0$ , the left side is $0^{n}$ . If $n = 0$ , both sides equal $1$ : the right side has the single empty tuple, coefficient $(\binom{0}{0}) = 1$ , and empty product $1$ . If $n \geq 1$ , the right side is an empty sum and the left side is $0$ ; this proves the formula in dimension zero.

1.3ih

Fix $m$ and assume the formula holds for this dimension for every exponent $j \in N$ and every $m$ -tuple of complex numbers.

2.1F2F7step 1.3given

Let $z_{0}, \dots, z_{m} \in C$ , put $S = \sum_{i < m} z_{i}$ , and fix $n \in N$ . The complex binomial theorem [F2], followed by the induction hypothesis [step 1.3] for each $j \leq n$ , expands $(S + z_{m})^{n}$ as the finite double sum over $0 \leq j \leq n$ and $β \in W (j, m)$ whose summand is $ι_{C} ⁣ ((\binom{n}{j})) ι_{C} ⁣ ((\binom{j}{β})) (\prod_{i < m} z_{i}^{β_{i}}) z_{m}^{n - j}$ .

3.1F1F3F4F5F6F7F9step 1.1step 2.1

For every pair $(j, β)$ in step 2.1, let $α = (β_{0}, \dots, β_{m - 1}, n - j) \in W (n, m + 1)$ . This is a bijection from the pair index set to $W (n, m + 1) :$ its inverse takes the first $m$ coordinates as $β$ and their natural sum as $j$ . Put $P = (\prod_{i < m} β_{i}!) (n - j)!$ . By [F3], $(\binom{n}{α}) P = n!$ ; also [F3] gives $(\binom{j}{β}) \prod_{i < m} β_{i}! = j!$ , so $(\binom{n}{j}) (\binom{j}{β}) P = (\binom{n}{j}) j! (n - j)! = n!$ . The factor $P$ is nonzero, since $(\binom{n}{α}) P = n! \neq 0$ by [F4]; hence [F5] gives $(\binom{n}{α}) = (\binom{n}{j}) (\binom{j}{β})$ . Step 1.1 carries this identity to the canonical naturals in $C$ , and the power laws [F6], [F9] identify the accompanying monomial with $\prod_{i < m + 1} z_{i}^{α_{i}}$ .

4.1F1F3F6F7F10step 1.1step 1.2step 1.3step 2.1step 3.1discharge-induction∎

Reindex the finite double sum of step 2.1 along the bijection in step 3.1. Its coefficients and monomials become exactly those in the asserted formula for $m + 1$ variables. The base case step 1.2 and this inductive step prove the statement for every $m$ by [F7] and induction. If all variables are zero and $n > 0$ , every $α \in W (n, m)$ has a positive coordinate, so every monomial on the right vanishes; if $n = 0$ , step 1.2 checks $0^{0} = 1$ . For $m = 1$ , the sole index is $(n)$ and its multinomial coefficient is $1$ by the coloring definition, so the formula reduces to $z_{0}^{n} = z_{0}^{n}$ .

Depends on

Used by

Bergman kernels of the disc, ball and polydisc, and Szegő kernels of the disc and ball Theorem

Dependency tree · two levels

66 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard P. Stanley, Enumerative Combinatorics, Volume 1, second edition (standard reference, not scraped)