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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 formula was first discovered by Abraham de Moivre [2] in the form ! [] +. De Moivre gave an approximate rational-number expression for the natural logarithm of the constant. Stirling's contribution consisted of showing that the constant is precisely 2 π {\displaystyle {\sqrt {2\pi }}} .
A related uniqueness theorem of Helmut Wielandt states that the complex gamma function and its scalar multiples are the only holomorphic functions on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2. [68] Other complex functions that interpolate the factorial ...
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.
De Polignac's formula; Difference operator; Difference polynomials; Digamma function; Egorychev method; Erdős–Ko–Rado theorem; Euler–Mascheroni constant; Faà di Bruno's formula; Factorial; Factorial moment; Factorial number system; Factorial prime; Gamma distribution; Gamma function; Gaussian binomial coefficient; Gould's sequence ...
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 ...
Faà di Bruno's formula; Factorial; Factorial moment; Factorial moment generating function; Factorial number system; Factorial prime; Fibonomial coefficient; Finite ...
This expression is found by writing 2 s Li s (−z) / (−z) = Φ(z 2, s, 1 ⁄ 2) − z Φ(z 2, s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e 2πt + 1) in place of 1 / (e 2πt − 1) to the second Φ series.