We consider the hyper-Bessel operator of order $$r\ge 2$$ : $$\begin{aligned} B_{\alpha }:= \frac{1}{z^{r-1}}\prod _{i=1}^{r-1} \big ( zD_z+({ r \alpha _{i} + 1})\big )D_z, \end{aligned}$$ where $$\alpha =(\alpha _1,\ldots ,\alpha _{r-1})$$ is a real multi-index such that $$\alpha _k \ge - 1 + {k}/{r}$$ for $$k=1,...,r - 1$$ and $$D_z$$ is the usual derivative in complex plane. We characterize the transmutation operators between two hyper-Bessel operators, namely from $$B_\beta $$ into $$B_\alpha $$ on the space $$H_r(\mathbb {C})$$ of r-even and entire functions with the help of the Sonine-Dimovski transform and we prove the spectral synthesis property associated with the operator $$B_\alpha $$ for the space $$H_r(\mathbb {C})$$ . Let us note that the hyper-Bessel operator $$B_{\alpha }$$ and the related transmutation operators can be also represented as operators of the generalized fractional calculus.