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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.
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