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