Definitions of factorials and binomials
The factorial , double factorial , Pochhammer symbol , binomial coefficient , and multinomial coefficient are defined by the following formulas. The first formula is a general definition for the complex arguments, and the second one is for positive integer arguments:
Remark about values at special points: For and integers with and, the Pochhammer symbol cannot be uniquely defined by a limiting procedure based on the previous definition because the two variables and can approach the integers and with and at different speeds. For such integers with , the following definition is used:
Similarly, for negative integers with , the binomial coefficient cannot be uniquely defined by a limiting procedure based on the previous definition because the two variables , can approach negative integers , with at different speeds. For negative integers with , the following definition is used:
The previous symbols are interconnected and belong to one group that can be called factorials and binomials. These symbols are widely used in the coefficients of series expansions for the majority of mathematical functions.
