Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let $\widehat{\mathfrak{g}}$ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type ${\widehat{BC}}_n=A_{2n}^{(2)}$). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with $\widehat{\mathfrak{g}}$. In type ${\widehat{A}}_{n-1}=A_{n-1}^{(1)}$ the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at $t=0$ specializes in turn to a well-known Pieri formula in the fusion ring of genus zero $\widehat{\mathfrak{s}\mathfrak{l}}(n{)}_c$-Wess-Zumino-Witten conformal field theories.