On Spherical Functions on the Group $\operatorname{SU}\times\operatorname{SU}\times\operatorname{SU}$

We consider the group $G:=\operatorname{SU}\times\dots\times\operatorname{SU}$ ($l$ factors), where $\operatorname{SU}(2)$ is the group of unitary $2\times 2$ matrices with unit determinant. Let $K\simeq\operatorname{SU}(2)$ be the diagonal subgroup of $G$. We obtain the generating function of all $K$-spherical functions on $G$.