Abstract
We consider the hyper-Bessel operator of order r >= 2:
B-alpha := 1/z(r-1) Pi(r-1)(i=1) (zD(z) + (r alpha(i) + 1)D-z,
where alpha = (alpha(1),..., alpha(r-1)) is a real multi-index such that alpha(k) >= -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 Ba on the space H-r (C) of r-even and entire functions with the help of the SonineDimovski transform and we prove the spectral synthesis property associated with the operator B-alpha for the space H-r (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.