Higher order logarithms of Bessel operators and an extension problem

We consider the Bessel operator defined by $$B_{\lambda }=-\frac{d^2}{d{x}^2}+\frac{{\lambda }^2-1\mathrm{/}4}{x^2},$$ on $(0,\mathrm{\infty })$, with $\lambda >-1$. We study the fractional power $B_{\lambda }^s$, $s\in (-1,1)$, $s\mathrm{\ne }0$, and the logarithm ${\log }^k{B}_{\lambda }$, $k\in \mathbb{N}$, of $B_{\lambda }$. We obtain pointwise representations of these operators and asymptotic Taylor expansions of the operators $B_{\lambda }^s$ in terms of logarithmic operators ${\log }^k{B}_{\lambda }$. We also obtain $\log B_{\lambda }$ as the solution of an extension problem.