  Classical integral representation of the Mellin type kernel in terms of the
Laplace integral gives an idea to construct a new class of non-convolution
(index) transforms. 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
