On $A^{\mathcal{I^{K}}}$–summability

In this paper, we introduce and investigate the concept of $A^{\mathcal{I^{K}}}$-summability as an extension of $A^{\mathcal{I^{*}}}$-summability which was recently (2021) introduced by O.H.H. Edely, where $A=(a_{nk})_{n,k=1}^{\infty}$ is a non-negative regular matrix and $\mathcal{I}$ and $\mathcal{K}$ represent two non-trivial admissible ideals in $\mathbb{N}$. We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that $A^{\mathcal{K}}$-summability always implies $A^{\mathcal{I^{K}}}$-summability whereas $A^{\mathcal{I}}$-summability not necessarily implies $A^{\mathcal{I^{K}}}$-summability. Finally, we give a condition namely $AP(\mathcal{I},\mathcal{K})$ (which is a natural generalization of the condition $AP$) under which $A^{\mathcal{I}}$-summability implies $A^{\mathcal{I^{K}}}$-summability.