Hadamard gamma

The Hadamard gamma function is defined by the formula $$H(x)=\dfrac{1}{\Gamma(1-x)} \dfrac{d}{dx} \log \left( \dfrac{\Gamma(\frac{1}{2}-\frac{x}{2})}{\Gamma(1-\frac{x}{2})} \right),$$ where $\Gamma$ denotes the gamma function.

Properties

Theorem: We can write $$H(x)=\dfrac{\psi(1-\frac{x}{2})-\psi(\frac{1}{2}-\frac{x}{2})}{2\Gamma(1-x)},$$ where $\psi$ is the digamma function.

Proof: proof goes here █

Theorem: The function $H$ is an entire function.

Proof: proof goes here █

Theorem: The function $H$ satisfies the formula $$H(x+1)=xH(x)+\dfrac{1}{\Gamma(1-x)}.$$

Proof: proof goes here █

References

Is the Gamma function misdefined?
Leonhard Euler's Integral: A Historical Profile of the Gamma Function
A superadditive property of Hadamard's gamma function