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$K$-SPHERICAL FUNCTIONS ON ABELIAN SEMIGROUPS
We present the form of the solutions 
$f:S\rightarrow \mathbb{C}$ of the functional equation where 
$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in K}f(x+\unicode[STIX]{x1D706}y)=|K|f(x)f(y)\quad \text{for }x,y\in S,\end{eqnarray}$$
$f$ satisfies the condition 
$f(\sum _{\unicode[STIX]{x1D706}\in K}\unicode[STIX]{x1D706}x)\neq 0$ for all 
$x\in S$, 
$(S,+)$ is an abelian semigroup and 
$K$ is a subgroup of the automorphism group of 
$S$.
