\section{Introduction}
Path integrals, Feynman diagrams and perturbation theory are well-established tools in the study of Quantum Field Theory (QFT). A typical introduction to these techniques is via "toy model" 
\begin{equation}
\label{eq:main_integral1}
Z(g) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty}  e^{- x^2 - g x^4} dx \;\;\; g \geq 0.
\end{equation}
The function $Z(g)$ is referred to as the partition function of i) {\it the zero dimensional anharmonic oscillator} \cite{HK} or alternatively ii) {\it zero dimensional $\phi^4$-field theory} \cite{JCC, Yuk}.  Analyzing the behaviour of (\ref{eq:main_integral1} is done via deriving perturbative expansions for $Z(g)$ in (\ref{eq:main_integral1}). This is done by expanding the exponential in the integral (\ref{eq:main_integral1}) and interchanging the order of the resulting summation and integration. In the {\it weak coupling} limit $g \rightarrow 0$ one obtains the divergent asymptotic expansion \cite{JCC, HK, Yuk}
\begin{equation}
\label{eq:series1}
Z(g) \sim \sum_{n=0}^{N} (-1)^n \frac{ \Gamma (2n + 1/2)}{n! \sqrt{\pi}} g^n.
\end{equation}
In the {\it strong coupling} limit $g \rightarrow \infty$ we obtain the convergent expansion  \cite{HK}
\begin{equation}
\label{eq:series2}
Z(g) \sim g^{-1/4} \sum_{n=0}^{N} (-1)^n \frac{ \Gamma (n/2 + 1/4)}{2n! \sqrt{\pi}} g^{-n/2}.
\end{equation}
The references \cite{HK, Yuk} state (without derivation) the closed-form solution to the integral (\ref{eq:main_integral1}):
\begin{equation}
\label{eq:main_soln1}
Z(g) = \frac{1}{\sqrt{4 \pi g}} \exp \left[\frac{1}{8g}\right]K_{1/4} \left( \frac{1}{8g} \right).
\end{equation} 
Note the {\it modified Bessel function of the second kind} $K_{\nu}$, also called a MacDonald function \cite{Yuk}, in (\ref{eq:main_soln1}). The function can be expressed for $|\arg z| < \pi/2$, i.e. Re $z > 0$, as \cite[p. 181]{GNW}:
\begin{equation}
\label{eq:KBess}
K_{\nu}(z) = \int_{0}^{\infty} \cosh (\nu t) e^{- z \cosh t} dt.
\end{equation}
The expression (\ref{eq:main_soln1} is in agreement with the formula in \cite[Ch. 3.323 No. 3 p. 337]{GRI}, which in turn refers to the formula in \cite[Ch. 4.5 No. 34 p. 147]{ERD}. No derivation of the formulae are stated in either \cite{ERD, GRI}. In addition we note the alternative formulation of the closed-form solution for $Z(g)$ in \cite{Yuk}:
\begin{equation}
\label{eq:main_soln5}
Z(g) = \left( \frac{1}{2g} \right)^{1/4} \exp \left[ \frac{1}{8g} \right] D_{-1/2} \left( \frac{1}{\sqrt{2g}} \right).
\end{equation}
The equation (\ref{eq:main_soln5}) can be verified from (\ref{eq:main_soln1}) using the identity \cite{Wolf}:
\begin{equation}
D_{-1/2}(z)=	\sqrt{\frac{z}{2 \pi}} \cdot K_{1/4} \left( \frac{z^2}{4} \right).
\end{equation}
As a final remark in this background section, we note that both expansions, (\ref{eq:series1}) and (\ref{eq:series2}), can also be obtained from (\ref{eq:main_soln1}) using the appropriate expansion of $K_{\nu}(z)$ \cite{Yuk}. 

\section{Derivation of Closed Form Expression for Partition Function}
We first prove the formula in \cite[Ch. 3.323 No. 3 p. 337]{GRI}. Using the same convention in \cite{GRI} we consider the integral
\begin{equation}
\label{eq:main_integral2}
I(\gamma, \beta) = \int_{0}^{\infty}  \exp \left[ - 2\gamma^2 x^2- \beta^2 x^4 \right] dx
\end{equation}
We are interested in the setting $\beta, \gamma \in \mathbb{R}$. We should remark that in the formula \cite[Ch. 3.323 No. 3 p. 337]{GRI} is quoted to hold under the more general conditions $| \arg \beta | < \pi/4$ and $| \arg \gamma | < \pi/2$. We define a new variable transformation
\begin{equation}
\label{eq:xtrans1}
x = \frac{\sqrt{2}\gamma}{\beta} \sinh \left( \frac{\xi}{4} \right).
\end{equation}
Using (\ref{eq:xtrans1}) (please refer details in Appendix \ref{sec:appA}) and the definition (\ref{eq:KBess}), we re-write ((\ref{eq:main_integral2}) as
\begin{eqnarray}
I(\gamma, \beta) &=& 2^{-3/2} \frac{\gamma}{\beta} \exp \left[ \frac{\gamma^4}{2\beta^2}  \right] \int_{0}^{\infty} \cosh \left( \frac{\xi}{4} \right) \exp \left[ - \frac{\gamma^4}{2\beta^2}  \cosh \xi \right] d\xi \nonumber \\
	&=& 2^{-3/2} \frac{\gamma}{\beta} \exp \left[ \frac{\gamma^4}{2\beta^2} \right] K_{1/4} \left( \frac{\gamma^4}{2\beta^2}  \right) \label{eq:main_soln2}
\end{eqnarray}
Equation ((\ref{eq:main_soln2}) is the formula  \cite[Ch. 3.323 No. 3 p. 337]{GRI}. To verify (\ref{eq:main_soln1}) we substitute $\gamma^2 = 1/2$ and $\beta^2 = g$ in (\ref{eq:main_soln2}) and use:
\begin{equation}
Z(g) = \frac{2}{\sqrt{\pi}} \cdot I\left(\pm \frac{1}{\sqrt{2}}, \pm \sqrt{g}\right).
\end{equation}

\section{Closed Form Expression for The Moments}
Let us return to partition function (\ref{eq:main_integral1}). We note that the integrand is an even function and should thus only have even (positive integer) moments given by the formula:
\begin{equation}
\label{eq:main_integral3}
\biggl< x^{2n} \biggr> = \frac{1}{Z(g)} \cdot \frac{2}{\sqrt{\pi}} \int_{0}^{\infty}  x^{2n} \exp \left[- x^2 - g x^4 \right] dx
\end{equation}
To obtain our closed form expression we proceed as before, now using the variable transformation
\begin{eqnarray}
\label{eq:xtrans4}
x &=& \frac{1}{ \sqrt{g} } \sinh \left( \frac{\xi}{4} \right) \\
\label{eq:xtrans5}
dx &=&  \frac{1}{ 4 \sqrt{g} } \cosh \left( \frac{\xi}{4} \right) d \xi.
\end{eqnarray}
Using  (\ref{eq:KBess}), (\ref{eq:xtrans4}) and (\ref{eq:xtrans5}) we re-write ((\ref{eq:main_integral3}), using the shorthand sh (ch) for sinh (cosh): 
\begin{equation}
\biggl< x^{2n} \biggr> = \mathcal{N} \int_0^{\infty} \mathrm{sh}^{2n} (\xi /4) \cdot \mathrm{ch}(\xi /4) \exp \left[ -\frac{1}{8g} \mathrm{ch}(\xi) \right] d\xi 
\label{eq:main_integral4}
\end{equation}
Note the normalization factor $\mathcal{N}=\mathcal{N}(n,g)$:
\[
 \mathcal{N} (n,g)= \frac{1}{Z(g)} \cdot \frac{e^{1/8g}}{\sqrt{4\pi g^{2n+1}}} =\left[ g^n \cdot K_{1/4} \left(\frac{1}{8g}\right) \right]^{-1}
\]
We now make use of
\[
\sinh^{2n} (x) = \left[ \cosh^2 (x) - 1 \right]^n = \sum_{k=0}^n  \binom{n}{k}(-1)^{n-k} \mathrm{ch}^{2k}(x) ,
\]
to transform (\ref{eq:main_integral4}) to
\begin{equation}
\label{eq:main_integral5}
\biggl< x^{2n} \biggr> = \mathcal{N}  \sum_{k=0}^{n}  \binom{n}{k}(-1)^{n-k} \int_0^{\infty} \mathrm{ch}^{2k+1}(\xi/4) \exp \left[ -\frac{1}{8g} \mathrm{ch}(\xi) \right] d\xi 
\end{equation}
We first use the trigonometric identity for odd powers of cosine \cite{Wolf2} and then apply Osborn's rule \cite{Wolf3} to convert the identity to the hyperbolic analogue:
\[
\cosh^{2k+1}(x) = \frac{1}{4^{k}} \sum_{m=0}^{k} \binom{2k+1}{m} \mathrm{ch}( (2k + 1 - 2m )\cdot x)
\]
Then we finish up (\ref{eq:main_integral5})
\begin{eqnarray}
\biggl< x^{2n} \biggr> &=& \mathcal{N} \sum_{k=0}^{n}  \binom{n}{k}\frac{(-1)^{n-k}}{4^{k}} \sum_{m=0}^{k}\binom{2k+1}{m} \times \nonumber \\
				& & \int_0^{\infty} \mathrm{ch}\left(\frac{(2(k-m)+1)\xi}{4}\right) \exp \left[ -\frac{1}{8g} \mathrm{ch}(\xi) \right] d\xi \nonumber \\
				&=& \mathcal{N} \sum_{k=0}^{n}  \binom{n}{k}\frac{(-1)^{n-k}}{4^{k}} \sum_{m=0}^{k} \binom{2k+1}{m} K_{\frac{2(k-m)+1}{4}} \left( \frac{1}{8g} \right) \label{eq:main_soln3}
\end{eqnarray}
In general we see that (\ref{eq:main_soln3}) is of the form
\begin{equation}
\biggl< x^{2n} \biggr> = \left[ (4g)^n \cdot K_{1/4}  \left( \frac{1}{8g} \right) \right]^{-1} \sum_{m=0}^n c_m^{(n)} \cdot  K_{\frac{2m+1}{4}},
\end{equation}
where with a little work we see that the coefficients $c_m^{(n)}$ can be expressed as: 
\begin{equation}
c_m^{(n)} = \sum_{k=m}^n (-1)^{n-k} 4^{n-k} \binom{n}{k} \binom{2k+1}{k-m}
\end{equation}
The coefficients can be simplified when starting from the top in descending order - for example the first few entries are
\begin{eqnarray}
c_n^{(n)} &=& 1 \\
c_{n-1}^{(n)}  &=& 1 - 2n \\
c_{n-2}^{(n)}  &=& n (2n-3) \\
c_{n-3}^{(n)}  &=& \frac{n (1 - 2n) (2n - 5)}{3}
\end{eqnarray}
The even moments ($2n$) for $n=1,2,3$ are then given by:
\begin{eqnarray}
\biggl< x^{2} \biggr> &=& \frac{1}{4g} \left[ \frac{K_{3/4} \left( \frac{1}{8g} \right)}{ K_{1/4}  \left( \frac{1}{8g} \right)} -1 \right] \\
\biggl< x^{4} \biggr> &=& \frac{1}{16g^2} \left[ \frac{K_{5/4} \left( \frac{1}{8g} \right)-3K_{3/4} \left( \frac{1}{8g} \right)}{ K_{1/4}  \left( \frac{1}{8g} \right)} +2 \right] \\
\biggl< x^{6} \biggr> &=& \frac{1}{64g^3} \left[ \frac{K_{7/4} \left( \frac{1}{8g} \right) -5 K_{5/4} \left( \frac{1}{8g} \right)+9K_{3/4} \left( \frac{1}{8g} \right)}{ K_{1/4}  \left( \frac{1}{8g} \right)} -5 \right] 
\end{eqnarray}

\section{Discussion}
As I was unable to find a derivation of the closed form expressions for $\phi^4$-field theory in zero dimensions, I set about deriving the expression on my own. Along the way the trick I used to derived the expression also enables one to write a closed form expression for the moments. I hope these results can lead to further insights on resummation methods used in the perturbative approach explored in \cite{JCC, HK, VM, Yuk} and references therein. It would be interesting to see if the hyperbolic transformation has a role to play in other field theories. I also discovered an erratum in one of the quoted formulas \cite{ERD} for which the correction is mentioned in the appendix. 


