Hankel type integral transforms connected with the hyper-Bessel differential operators

In this paper we give a solution of a problem posed by the second author in her book, namely, to find symmetrical integral transforms of Fourier type, generalizing the cos-Fourier (sin-Fourier) transform and the Hankel transform, and suitable for dealing with the hyper-Bessel differential operators of order m>1 ${\displaystyle B:=x^{-\beta }{\prod }_{j=1}^m(x\left(\frac{d}{dx}\right)+\beta {\gamma }_j)}$, β>0, ${\displaystyle {\gamma }_j\in R}$, j=1,...,m. We obtain such integral transforms corresponding to hyper-Bessel operators of even order 2m and belonging to the class of the Mellin convolution type transforms with the Meijer G-function as kernels. Inversion formulas and some operational relations for these transforms are found.