I'm looking to compute an exact integral formula for the reciprocal of the double factorial function, $(2n-1)!!$, or just as easily for the reciprocal gamma function, $\Gamma\left(n+\frac{1}{2}\right)$. I found the post located here and that formula works well for me when, for example, I take $c := 1$.
However, there is another known formula that I'm looking to replicate, or at least find a suitable analog to. Namely, that for integers $n \geq 0$ we have that (this formula is found in the appendices of the Concrete Mathematics book, for example):
\begin{align*}
\frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-n\imath t} e^{e^{\imath t}} dt & = \frac{1}{n!}.
\end{align*}
I have had a look around google and found page 3 of this article and the Hankel loop contour described here, though I am still struggling to find the analog to this formula for the double factorial function case. I believe that the integral formula above is derived from the contour integral for the reciprocal gamma function, but when I perform a change of variable in the loop contour formula and plugin $z \mapsto n + \frac{1}{2}$, Mathematica computes the following result when $n = 3$ (the expected result is acceptably $\frac{8}{105}$, or ideally $\frac{1}{105}$):
\begin{align*}
\frac{1}{4\sqrt{\pi}} \int_{-\pi}^{\pi} e^{-(n_3+\frac{1}{2})\imath t} e^{e^{\imath t}} dt & = -\frac{1}{21 e \sqrt{\pi}} + \frac{8}{105} \operatorname{erf}(1).
\end{align*}
The result is obviously close to the intended formula, so I'm thinking that perhaps it's an issue with the bounds on the integral. I would like to keep the bounds of integration finite as in the factorial function formula if possible. Does anyone have any thoughts, advice, or solutions for this problem?
Best Answer
Update: The solution to my original question, which was asking if given an OGF $F(z)$ for some sequence $\{f_n\}_{n \geq 0}$, whether there is an integral transform that generalizes the known OGF-to-EGF transform given by $$\widehat{F}(z) = \frac{1}{2\pi} \int_{-\pi}^{\pi} F\left(z e^{-\imath t}\right) e^{e^{\imath t}} dt,$$ can be answered with Fourier series and integral representations for the Hadamard product of two generating functions. I'm actually writing up a short note one this now, but the integral transform in the previous question is given by $$\sum_{n \geq 0} \frac{f_n z^n}{(2n+1)!!} = \frac{1}{2\sqrt{2\pi}} \int_{-\pi}^{\pi} F\left(z e^{-\imath t}\right) e^{\frac{1}{2}\left(e^{\imath t} -\imath t\right)} \operatorname{erf}\left(\frac{e^{\imath t/2}}{\sqrt{2}}\right) dt.$$