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In the same way that the double factorial generalizes the notion of the single factorial, the following definition of the integer-valued multiple factorial functions (multifactorials), or α-factorial functions, extends the notion of the double factorial function for positive integers : ! = {()!
The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, [18] in the first work on Faà di Bruno's formula, [19] but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial. [20]
In the previous two integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.
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 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
One particular formula results in the case of the double factorial function example given immediately below in this section. The last integral formula is compared to Hankel's loop integral for the reciprocal gamma function applied termwise to the power series for F ( z ) {\displaystyle F(z)} .
This formula can be motivated from the combinatorial definition and thus serves as a natural starting point for the theory. For small values of n {\textstyle n} and k {\textstyle k} , the values of A ( n , k ) {\textstyle A(n,k)} can be calculated by hand.
The probability density function of the Gaussian q-distribution is given by = ... [2n − 1]!! is the q-analogue of the double factorial given by ...