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[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [ 41 ] Factorials are used extensively in probability theory , for instance in the Poisson distribution [ 42 ] and in the probabilities of random permutations . [ 43 ]
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
This is a list of factorial and binomial topics in mathematics. ... Factorial; Factorial moment; Factorial number system; Factorial prime; Gamma distribution;
Let be a natural number. For a base >, we define the sum of the factorials of the digits [5] [6] of , :, to be the following: = =!. where = ⌊ ⌋ + is the number of digits in the number in base , ! is the factorial of and
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
This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms. Exponential function [ edit ]
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