Is there a way to indicate $NP^{\prod_i^P}$ or $coNP^{\prod_i^P}$ like in polynomial hierarchy?

Question

In know that in polynomial hierarchy

• $\ \ \ \sum_{i+1}^P=NP^{\sum_i^P}$
• $\ \ \ \prod_{i+1}^P=coNP^{\sum_i^P}$

I was wondering if there exists a way (and if it does make sense) to indicate $NP^{\prod_i^P}$ or $coNP^{\prod_i^P}$.