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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]
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
(Here we use the standard notations and conventions of lambda calculus: Y is a function that takes one argument f and returns the entire expression following the first period; the expression . ( ) denotes a function that takes one argument x, thought of as a function, and returns the expression ( ), where ( ) denotes x applied to itself ...
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.
These are counted by the double factorial 15 = (6 − 1)‼. In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. [1] That is,
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. 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.
The math template formats mathematical formulas generated using HTML or wiki markup. (It does not accept the AMS-LaTeX markup that <math> does.) The template uses the texhtml class by default for inline text style formulas, which aims to match the size of the serif font with the surrounding sans-serif font (see below).
When an inline formula is long enough, it can be helpful to allow it to break across lines. Whether using LaTeX or templates, split the formula at each acceptable breakpoint into separate <math> tags or {} templates with any binary relations or operators and intermediate whitespace included at the trailing rather than leading end of a part.