Multinomial theorem

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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 .


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 .


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

for everyone with ,



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


Web links