A generalization of both the gamma and inverse-gamma distribution is the generalized inverse-gamma distribution, which has density

\[
\begin{equation*}
f(s)=\frac{(a / b)^{p / 2}}{2 K_{p}(\sqrt{a b})} s^{p-1} \mathrm{e}^{-\frac{1}{2}(a s+b / s)}, \quad a, b, s>0, \quad p \in \mathbb{R} \tag{4.48}
\end{equation*}
\]

where \(K_{p}\) is the modified Bessel function of the second kind, which can be defined as the integral

\[
\begin{equation*}
K_{p}(x)=\int_{0}^{\infty} \mathrm{e}^{-x \cosh (t)} \cosh (p t) \mathrm{d} t, \quad x>0, p \in \mathbb{R} \tag{4.49}
\end{equation*}
\]

We write \(S \sim \operatorname{GIG}(a, b, p)\) to denote that \(S\) has a pdf of the form (4.48). The function \(K_{p}\) has many interesting properties. Special cases include

\[
\begin{aligned}
K_{1 / 2}(x) & =\sqrt{\frac{x \pi}{2}} \mathrm{e}^{-x} \frac{1}{x} \\
K_{3 / 2}(x) & =\sqrt{\frac{x \pi}{2}} \mathrm{e}^{-x}\left(\frac{1}{x}+\frac{1}{x^{2}}\right) \\
K_{5 / 2}(x) & =\sqrt{\frac{x \pi}{2}} \mathrm{e}^{-x}\left(\frac{1}{x}+\frac{3}{x^{2}}+\frac{3}{x^{3}}\right)
\end{aligned}
\]

More generally, \(K_{p}\) satisfies the recursion

\[
\begin{equation*}
K_{p+1}(x)=K_{p-1}(x)+\frac{2 p}{x} K_{p}(x) \tag{4.50}
\end{equation*}
\]

(a) Using the change of variables \(\mathrm{e}^{z}=s \sqrt{a / b}\), show that

\[
\int_{0}^{\infty} s^{p-1} \mathrm{e}^{-\frac{1}{2}(a s+b / s)} \mathrm{d} s=2 K_{p}(\sqrt{a b})(b / a)^{p / 2}
\]

(b) Let \(S \sim \operatorname{GIG}(a, b, p)\). Show that

\[
\begin{equation*}
\mathbb{E} S=\frac{\sqrt{b} K_{p+1}(\sqrt{a b})}{\sqrt{a} K_{p}(\sqrt{a b})} \tag{4.51}
\end{equation*}
\]

and

\[
\begin{equation*}
\mathbb{E} S^{-1}=\frac{\sqrt{a} K_{p+1}(\sqrt{a b})}{\sqrt{b} K_{p}(\sqrt{a b})}-\frac{2 p}{b} \tag{4.52}
\end{equation*}
\]