Laplace Transform of $\frac{\psi ^{(2)}\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}$?

Question

Is there a closed form expression for the Laplace Transform of the following expression?

$$f(x)=\frac{\psi ^{(2)}\left(\frac{1}{\sqrt{x}}\right)}{x^{3/2}}$$

where $\psi^{(k)}$ is the polygamma function.

$f(x)$ arises when sampling numbers $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\ldots$ with relative frequencies $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\ldots$, and treating this distribution as continuous. $f(x)$ is the probability density function.

Answer 1

Not an explicit expression, but a useful asymptotic expansion for large $z$ may be obtained using $(5.15.9)$ and Watson's lemma: \begin{align*} \int_0^{ + \infty } {{\rm e}^{ - zt} t^{ - 3/2} \psi ^{(2)} (t^{ - 1/2} )\,{\rm d}t} & \sim - \int_0^{ + \infty } {{\rm e}^{ - zt} t^{ - 3/2} \bigg( {t + t^{3/2} + \sum\limits_{k = 1}^\infty {(2k + 1)B_{2k} t^{k + 1} } } \bigg){\rm d}t} \\ & \sim - \sqrt {\frac{\pi }{z}} - \frac{1}{z} - 2\sum\limits_{k = 1}^\infty {\frac{{\Gamma\! \left( {k + \frac{3}{2}} \right)\!B_{2k} }}{{z^{k + 1/2} }}} , \end{align*} as $z\to \infty$ in the sector $|\arg z|\le \frac{\pi}{2}-\delta< \frac{\pi}{2}$. The sector of validity may be enlarged and explicit error bounds can be given if necessary.