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One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same functional equation.
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. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:
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 n ! ! = n × ( n − 2 ) ! ! {\displaystyle n!!=n ...
In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers.Derived by Daniel Bernoulli, the gamma function () is defined for all complex numbers except non-positive integers, and for every positive integer =, () = ()!.
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
Special cases of the Pochhammer k-symbol, (),, correspond to the following special cases of the falling and rising factorials, including the Pochhammer symbol, and the generalized cases of the multiple factorial functions (multifactorial functions), or the -factorial functions studied in the last two references by Schmidt:
Consider the factorial function F(n) recursively defined by F(n) = 1, if n = 0; else n × F(n − 1). In the lambda expression which is to represent this function, a parameter (typically the first one) will be assumed to receive the lambda expression itself as its value, so that calling it – applying it to an argument – will amount to ...
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Li s (z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function .