Multinomial theorem

In mathematics , the multinomial theorem (also multinomial formula or multinomial theorem ) or polynomial theorem represents a generalization of the binomial formula to the sum of any number of coefficients by generalizing the binomial coefficients as multinomial coefficients .

formula

The multinomial coefficient is for nonnegative integers and is defined as ${\ displaystyle k_ {1}, \ ldots, k_ {n}}$ ${\ displaystyle k: = \! \, k_ {1} + \ ldots + k_ {n}}$

{\ displaystyle {k \ choose k_ {1}, \, \ ldots, \, k_ {n}}: = {\ frac {k!} {k_ {1}! \ cdot \, \ ldots \, \ cdot k_ {n}!}}.}

The multinomial theorem is then

{\ displaystyle (x_ {1} + x_ {2} + \ ldots + x_ {n}) ^ {k} \, = \ sum _ {k_ {1} + \ ldots + k_ {n} = k} {k \ choose k_ {1}, \ ldots, k_ {n}} \, \ cdot \, x_ {1} ^ {k_ {1}} \ cdot x_ {2} ^ {k_ {2}} \ cdots x_ {n } ^ {k_ {n}}.}

The multi-index notation with multi-index allows a shorter formulation : ${\ displaystyle \ alpha}$

{\ displaystyle (x_ {1} + x_ {2} + \ cdots + x_ {n}) ^ {k} = \ sum _ {| \ alpha | = k} {{k} \ choose \ alpha} \ cdot x ^ {\ alpha}}

One identifies with the vector . ${\ displaystyle x}$ ${\ displaystyle (x_ {1}, \ ldots, x_ {n}) \ in \ mathbb {R} ^ {n}}$

application

As a corollary from the multinomial theorem, for example, one obtains the estimate for multi-indices

{\ displaystyle n ^ {k} = (1+ \ cdots +1) ^ {k} = \ sum _ {| \ beta | = k} {\ frac {| \ beta |!} {\ beta!}} \ geq {\ frac {| \ alpha |!} {\ alpha!}}}

for everyone with ,

{\ displaystyle \ alpha}

{\ displaystyle | \ alpha | = k}

so

{\ displaystyle | \ alpha |! \ leq n ^ {| \ alpha |} \ cdot \ alpha!}

.

Evidence sketch

The multinomial theorem can be either with the help of a multi-dimensional Taylor expansion of the first order or by induction through the aid of the binomial theorem proving. ${\ displaystyle n}$

literature

SA Rukova: Multinomial coefficient . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Jaroslav Nesetril, Jiri Matousek: Discrete Mathematics: A Journey of Discovery . Springer 2007, ISBN 978-3-540-30150-9 , p. 79 ( excerpt in the Google book search)

Dominique Foata, Aimé Fuchs: Probability Theory . Birkhäuser 1999, ISBN 3-7643-6169-7 , pp. 41-42 ( excerpt in the Google book search)

Web links

Eric W. Weisstein : Multinomial Coefficient . In: MathWorld (English).