  We establish new characterizations of the Bloch space $\mathcal{B}$ which
include
  descriptions in terms of classical fractional derivatives.
  Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty
\widehat{f}(n) z^n$ in the unit disc $\mathbb{D}$,
  we define the fractional derivative
  $
  D^{\mu}(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{\mu_{2n+1}}
z^n
  $ induced by a radial weight $\mu$,
  where $\mu_{2n+1}=\int_0^1 r^{2n+1}\mu(r)\,dr$ are the odd moments of $\mu$.
Then, we consider
  the space $
  \mathcal{B}^\mu$ of analytic functions $f$ in $\mathbb{D}$ such that
$\|f\|_{\mathcal{B}^\mu}=\sup_{z\in \mathbb{D}}
\widehat{\mu}(z)|D^\mu(f)(z)|<\infty$, where $\widehat{\mu}(z)=\int_{|z|}^1
\mu(s)\,ds$.
  We prove that $\mathcal{B}^\mu$ is continously embedded in $\mathcal{B}$ for
any radial weight $\mu$, and $\mathcal{B}=\mathcal{B}^\mu$ if and only if
$\mu\in \mathcal{D}=\widehat{\mathcal{D}}\cap\check{\mathcal{D}}$. A radial
weight $\mu \in \widehat{\mathcal{D}}$ if $\sup_{0\le r
<1}\frac{\widehat{\mu}(r)}{\widehat{\mu}\left(\frac{1+r}{2}\right)}<\infty$ and
a radial weight $\mu \in \check{\mathcal{D}}$ if there exist $K=K(\mu)>1$ such
that $\inf_{0\le r<1}\frac{\widehat{\mu}(r)}{
\widehat{\mu}\left(1-\frac{1-r}{K}\right)}>1.$
