# Relation between Gamma function and base 2 exponential

In my work, I keep coming across the term

$$f(x) = \frac{1}{\sqrt{2\pi}} 2^{\frac{x-1}{2}} \Gamma(\frac{x+1}{2}),$$

in particular for $$x \in [0,1]$$. I have been working on bounding the density function of a product of powers of standard Gaussians, i.e.

$$Y = X_1^{e_1}\cdots X_k^{e_k}$$

for $$X_i \sim \mathcal{N}(0,1)$$ i.i.d., and have been applying those bounds to bound e.g.

$$E[\lvert Y \rvert^{\delta}],$$

for $$\delta \in [0,1]$$.

Has anyone else come across this kind of term before? And could you help me understand/prove why, in the range of $$x \in [0,1]$$, it is bounded by $$\frac{1}{2}$$?

Thank you!

Edit: Added more detail, clarified the question, added the missing constant. Thank you for the feedback.

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Sep 14 at 16:29
• This should be pretty close to $n!!$, the double factorial. Sep 14 at 17:23
• I have edited the question to be more specific. Thank you for your feedback. Sep 14 at 23:23
• I cannot help you understand why $f(x)$ is bounded by 1/2 for $x\in [0,1]$. The problem is it does not seem to be so. For instance, I find $f(0)=\sqrt{\pi/2}$. Sep 15 at 2:36
• @MichaelRenardy The usual problem, I forgot that I always carry the constant $\frac{1}{\sqrt{2\pi}}$. Sloppiness on my part. Sep 15 at 11:09