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
The multinomial theorem is then
The multi-index notation with multi-index allows a shorter formulation :
One identifies with the vector .
application
As a corollary from the multinomial theorem, for example, one obtains the estimate for multi-indices
- for everyone with ,
so
- .
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.
See also
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).