A class of index transforms generated by the Mellin and Laplace operators

Classical integral representation of the Mellin type kernel x − z = 1 Γ ( z ) ∫ 0 ∞ e − x t t z − 1 d t , x > 0 , Re z > 0 , in terms of the Laplace integral gives an idea to construct a class of non-convolution (index) transforms with the kernel k z ± ( x ) = ∫ 0 ∞ e − x t ± 1 r ( t ) t z − 1 d t , x > 0 , where r ( t ) ≠ 0 , t ∈ R + admits a power series expansion, which has an infinite radius of convergence and the integral converges absolutely in a half-plane of the complex plane z . Particular examples give the Kontorovich–Lebedev-like transformation and new transformations with hypergeometric functions as kernels. Mapping properties and inversion formulas are obtained. Finally we prove a new inversion theorem for the modified Kontorovich–Lebedev transform.