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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 : ! = {()!
Definition The hyperfactorial of ... can be continuously interpolated by the gamma function, ... where !! is the notation for the double factorial. [4] The ...
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 ...
A corresponding relation holds for the rising factorial and the backward difference operator. The study of analogies of this type is known as umbral calculus. A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences ...
"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." [1]: 38 The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. [1]
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
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
Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function. [ 2 ] According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p {\displaystyle p} is an odd prime number s f ( p − 1 ) ≡ ( p − 1 ) ! !