The heat kernel associated with generalized Kontorovich–Lebedev transform

We study some properties of the eigenfunction φα,iτμ,β of the following second-order differential operator defined on (0,∞), by Δx,αμ,β=x2d2dx2+(1−2α)xddx−β2μ2x2μ;α,β,μ∈(0,∞). We obtain estimates of the integral of the kernel wα,β,μ, translation and convolution product related to φα,iτμ,β. We define and explore a generalization of the Kontorovich–Lebedev transform. The heat kernel and Weierstrass transform tied to Δx,αμ,β are considered.