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Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients , double factorials , falling factorials ...
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre . It is also sometimes known as de Polignac's formula , after Alphonse de Polignac .
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.
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations.It is also called factorial base, although factorials do not function as base, but as place value of digits.
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation!! = ()!! to give !! = (+)!! +.
2.4 Modified-factorial denominators. ... is a Bernoulli number, and here, = . is an Euler number ... is the Riemann zeta function.
The falling factorial can be extended to real values of using the gamma function provided and + are real numbers that are not negative integers: = (+) (+) , and so can the rising factorial: = (+) . Calculus
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]