ON \(A^{\mathcal{I^{K}}}\)–SUMMABILITY

Chiranjib Choudhury     (Tripura University (A Central University), Suryamaninagar-799022, Agartala, India)
Shyamal Debnath     (Tripura University (A Central University), Suryamaninagar-799022, Agartala, India)


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.


Ideal, Filter, \(\mathcal{I}\)-convergence, \(\mathcal{I^{K}}\)-convergence, \(A^{\mathcal{I}}\)-summability, \(A^{\mathcal{I^{K}}}\)-summability.

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DOI: http://dx.doi.org/10.15826/umj.2022.1.002

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