mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a pseudogamma function is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

that interpolates the

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

. The

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of curves can be drawn through those points. Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function. The two most famous pseudogamma functions are

Hadamard's gamma function In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function). This function, with its argument shift ...

H(x)=\frac = \frac

where

\Phi

is the

Lerch zeta function In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887. The Ler ...

, and the Luschny factorial:

\Gamma(x+1)\left(1-\frac\left(\frac\left(\psi\left(\frac\right)-\psi\left(\frac\right)\right)-\frac\right)\right)

where denotes the classical gamma function and denotes the

digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...

. Other related pseudogamma functions are also known. However, by adding conditions to the function interpolating the factorial, we obtain uniqueness of this function, most often given by the Gamma function. The most common condition is the

logarithmic convexity In mathematics, a function ''f'' is logarithmically convex or superconvex if \circ f, the composition of the logarithm with ''f'', is itself a convex function. Definition Let be a convex subset of a real vector space, and let be a function tak ...

: this is the Bohr-Mollerup theorem. See also the

Wielandt theorem In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers z for which \mathrm\,z > 0 by :\Gamma(z)=\int_0^ t^ \mathrm e^\,\mathrm dt, as the only function f defined on the half-plane H := \ such that: * ...

for other conditions.

References

{{Mathanalysis-stub Functions and mappings Factorial and binomial topics