Let \(b\ge 2\) , \(q>p\ge 2\) be three integers and let \(\mathcal {D} =\mathcal {D} _b \oplus b^{p}\mathcal {D} _b\) be a product-form digit set, where \(\mathcal {D}_b=\{0,1, \ldots , b-1\}\) . As is well known, the self-similar measure \(\mu _{b^q, \mathcal {D}}\) induced by the pair \((b^q,\mathcal {D})\) is a spectral measure, and it has a spectrum \( \Lambda \left( b^q, \mathcal {C}\right) =\left\{ \sum _{i=0}^{\text{ n }} c_{i} b^{q i}: n\in \mathbb {N}~\text {and}~ c_{i} \in \mathcal {C}=b^{q-p-1}\mathcal {D}\right\} . \) That is, the exponential function family \(E(\Lambda ):=\left\{ e^{2 \pi i \lambda x}: \lambda \in \Lambda \right\} \) forms an orthonormal basis in the Hilbert space \(L^2(\mu _{b^q, \mathcal {D}})\) . In this paper, for pairwise distinct primes \(t_1,t_2,\ldots , t_n\) and arbitrary non-negative integers \(k_{1},k_{2}\ldots , k_n\) , we give some sufficient conditions such that the exponential function family \(E(\prod _{i=1}^{n} t_{i}^{k_{i}}\Lambda )\) also forms an orthonormal basis in \(L^2(\mu _{b^q, \mathcal {D}})\) , and we will provide examples to illustrate these results.
