How to treat negative integer factorial function?

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Negative integer factorial functions do not have a standard expression, as they are not defined in the conventional sense. The Gamma function serves as an analytic continuation of the factorial function but has poles at negative integers, complicating its use for this purpose. To address negative factorials, one must first establish a clear definition of what a "negative factorial function" entails. The discussion emphasizes the need for context, particularly if the concept was introduced in a specific text or exercise. Ultimately, without a defined framework, negative integer factorials remain an ambiguous topic in mathematics.
dongsh2
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Dear all,

How to deal with negative integer factorial functions? I mean what expression formula can be substituted for this?
 
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I don't believe there is one. The Gamma function is the analytic continuation, but it has poles at the negative integers.
 
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The first thing you would have to do is define this "negative factorial function". Where did you find a reference to a "negative factorial function"? If this was given in some text exercise how was the negative factorial function defined?
 

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