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Let $$b_{k}(n)$$ bk(n) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for $$b_{7}(n)$$ b7(n) and $$b_{49}(n)$$ b49(n) . For example, for all $$j\ge 1$$ j≥1 and $$n\ge 0$$ n≥0 , we prove that $$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$ b7(72j-1n+3·72j-1-14)≡0(mod7j)...
In this paper, we establish several modular relations for the Rogers–Ramanujan type functions of order eleven which are analogous to Ramanujan's forty identities for Rogers–Ramanujan functions. Furthermore, we give interesting partition-theoretic interpretation of some of the modular relations which are derived in this paper.
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