Explicit fundamental solution for the operator $L+\alpha|T|$ on the Gelfand pair $(\mathbb{H}_{n},U(n))$

By means of the spherical functions associated to the Gelfand pair $(\mathbb{H}_{n},U(n))$ we define the operator $L+\alpha |T|$, where $L$ denotes the Heisenberg sublaplacian and $T$ denotes de central element of the Heisenberg Lie algebra, we establish a notion of fundamental solution and explicitly compute in terms of the Gauss hypergeometric function. For $\alpha<n$ we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when $\alpha=0$ we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.