The simplest definition of binomial coefficients is given by

where *n* is a positive integer and 0 ≤ *k* ≤ *n*. There are two generalizations of this definition. The first is to let n be any real number. Then the binomial coefficient (*r*, *k*) can be defined as

where

is the *k*th falling power of *r*. This definition is adequate for most applications.

The final generalization is to let *z* and *w* be any complex numbers and define the binomial coefficient (*z*, *w*) as

This generalization is more complicated, but in some sense it is more natural. Essentially it uses the simplest definition and replaces *n*! with Γ(*n*+1). However, the business of the limits is subtle and important.

For motivation of these definitions and details regarding how they work, see the article Binomial coefficients.

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