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Congruences modulo 8 for \((2,\, k)\) -regular overpartitions for odd \(k > 1\)

In this paper, we study various arithmetic properties of the function \(\overline{p}_{2,\,\, k}(n)\), which denotes the number of \((2,\,\, k)\)-regular overpartitions of n with odd \(k > 1\). We prove several infinite families of congruences modulo 8 for \(\overline{p}_{2,\,\, k}(n)\). For example, we find that for all non-negative integers \(\beta , n\) and \(k\equiv 1\pmod {8...