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 than Euler's gamma function. It is defined as:


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

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


 * $$H(n) = \Gamma(n) = (n-1)! $$

Properties
Unlike the classical gamma function, Hadamard's gamma function $H(x)$ is an entire function, i.e. it has no poles in its domain. It satisfies the functional equation


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

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

Representations
Hadamard's gamma can also be expressed as


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

where $$\Phi$$ is the Lerch zeta function, and as


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

where $ψ(x)$ denotes the digamma function.