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The factorial function of a positive integer ... According to this formula, the number of zeros can be obtained by subtracting the base-5 digits of ...
However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
The rising factorial is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for | | < by the power series (,;;) = = () ()! provided that ,,, …. Note, however, that the hypergeometric function literature typically uses the notation ( a ) n {\displaystyle (a)_{n}} for rising factorials.
The term odd factorial is sometimes used for the double factorial of an odd number. [ 5 ] [ 6 ] The term semifactorial is also used by Knuth as a synonym of double factorial.
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120. By convention, the value of 0! is defined as 1. This classical factorial function appears prominently in many theorems in number theory. The following are a few of these theorems. [1]
More recent results providing Jacobi-type J-fractions that generate the single factorial function and generalized factorial-related products lead to other new congruence results for the Stirling numbers of the first kind. [13] For example, working modulo we can prove that
A natural number is a sociable factorion if it is a periodic point for , where = for a positive integer, and forms a cycle of period . A factorion is a sociable factorion with k = 1 {\displaystyle k=1} , and a amicable factorion is a sociable factorion with k = 2 {\displaystyle k=2} .
Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive integer then 4 divides ( 2 n n ) {\displaystyle {\binom {2n}{n}}} if and only if n is not a power of 2.
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