\begin{theorem}
The reciprocal of 
assumes the following family of integral representations
(1) ![\begin{equation*} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+1}}{(2m)!\, 2^{4m+2n}} \int_0^{\infty} \frac{x^{2m} J_{2m+1}(\lambda x)}{\Gamma\left(m+ n+\frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation*}](https://quant-math.org/wp/wp-content/ql-cache/quicklatex.com-1d533d9fd746df619efadcb08cecbbc4_l3.png "Rendered by QuickLaTeX.com")
(2) ![\begin{equation*} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+2}}{(2m+1)!\, 2^{4m+2n+1}} \int_0^{\infty} \frac{x^{2m+1} J_{2m+2}(\lambda x)}{\Gamma\left(m+n+ \frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation*}](https://quant-math.org/wp/wp-content/ql-cache/quicklatex.com-3c43ff413811c7f9a81c20f9037576ff_l3.png "Rendered by QuickLaTeX.com")
for 
, 
and 
, where 
is the Bessel function of the first kind and 
is the Euler gamma function.
\end{theorem}
\begin{theorem}
The reciprocal of assumes the following family of integral representations
(1)
(2)
for , and , where is the Bessel function of the first kind and is the Euler gamma function.
\end{theorem}