Sonine-Dimovski transform and spectral synthesis associated with the hyper-Bessel operator on the complex plane

Abstract

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.