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. This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way from Euler's gamma function. It is defined as:
where denotes the classical gamma function. If is a positive integer, then:

Properties

Unlike the classical gamma function, Hadamard's gamma function is an entire function, i.e., it is defined and analytic at all complex numbers. It satisfies the functional equation
with the understanding that is taken to be for positive integer values of.
The Hadamard's gamma function has a superadditive property:
for all, where is the unique solution to the equation in the interval.

Representations

Hadamard's gamma can also be expressed as
and also as
where denotes the digamma function, and denotes the Lerch zeta function.