The heat kernel associated with generalized Kontorovich–Lebedev transform

Abstract

ABSTRACT We study some properties of the eigenfunction φ α , iτ μ , β of the following second-order differential operator defined on ( 0 , ∞ ) , by Δ x , α μ , β = x 2 d 2 d x 2 + ( 1 − 2 α ) x d d x − β 2 μ 2 x 2 μ ; α , β , μ ∈ ( 0 , ∞ ) . We obtain estimates of the integral of the kernel w α , β , μ , translation and convolution product related to φ α , iτ μ , β . We define and explore a generalization of the Kontorovich–Lebedev transform. The heat kernel and Weierstrass transform tied to Δ x , α μ , β are considered.