Hadamard gamma

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

Theorem: Define $\alpha_0 \approx 1.5031\ldots$ to be the only solution of the equation $H(2t)=2H(t)$. Then the inequality $H(x)+H(y) \leq H(x+y)$ holds for all real numbers $x,y$ with $x,y \geq \alpha$ if and only if $\alpha \geq \alpha_0$.

Proof: proof goes here █

See Also

Gamma

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