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

Abstract

Classical integral representation of the Mellin type kernel x−z=1Γ(z)∫0∞e−xttz−1dt,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 kz±(x)=∫0∞e−xt±1r(t)tz−1dt,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.