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 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:

${\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma (1-{\frac {x}{2}})}}\right)\right\},}$

where Γ(x) denotes the classical gamma function. If n is a positive integer, then:

${\displaystyle H(n)=\Gamma (n)=(n-1)!}$

Unlike the classical gamma function, Hadamard's gamma function H(x) is an entire function, i.e., it is defined and analytic at all complex numbers. It satisfies the functional equation

${\displaystyle H(x+1)=xH(x)+{\frac {1}{\Gamma (1-x)}},}$

with the understanding that ${\displaystyle {\tfrac {1}{\Gamma (1-x)}}}$ is taken to be 0 for positive integer values of x.

The Hadamard's gamma function has a superadditive property:

${\displaystyle H(x)+H(y)\leq H(x+y),}$

for all ${\displaystyle x,y\geq \alpha }$, where ${\displaystyle \alpha =1.5031...}$ is the unique solution to the equation ${\displaystyle H(2t)=2H(t)}$ in the interval ${\displaystyle [1.5,\infty )}$.¹

Hadamard's gamma can also be expressed as

${\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\Gamma (1-x)}}={\frac {L\left(-1,1,-x\right)}{\Gamma (-x)}},}$

and also as

${\displaystyle H(x)=\Gamma (x)\left[1+{\frac {\sin(\pi x)}{2\pi }}\left\{\psi \left({\dfrac {x}{2}}\right)-\psi \left({\dfrac {x+1}{2}}\right)\right\}\right],}$

where ψ(x) denotes the digamma function, and ${\displaystyle L}$ denotes the Lerch zeta function.