Journal of the Ramanujan Mathematical Society

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New Infinite Families of Congruences Modulo 3, 5 and 7 For Overpartition Function


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1 Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791 112, Arunachal Pradesh, India
     

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Let ¯p(n) denote the number of overpartitions of a non-negative integer n. In this paper, we prove two new infinite families of congruences modulo 3 for ¯p(n) by using Ramanujan’s theta-function identities. Particularly, we prove that, for any integer α ≥ 0, ¯p(9α+1(24n + 23)) ≡ 0 (mod 3) and ¯p(9α+1(24n + 22) + 1) ≡ 0 (mod 3). Furthermore, we prove some new congruences modulo 5 and 7 for ¯p(n). For example, we prove that ¯p(5n+k+3) ≡ 0 (mod 5), where k = 3n2 ± n.
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  • New Infinite Families of Congruences Modulo 3, 5 and 7 For Overpartition Function

Abstract Views: 485  |  PDF Views: 1

Authors

Jubaraj Chetry
Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791 112, Arunachal Pradesh, India
Nipen Saikia
Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791 112, Arunachal Pradesh, India

Abstract


Let ¯p(n) denote the number of overpartitions of a non-negative integer n. In this paper, we prove two new infinite families of congruences modulo 3 for ¯p(n) by using Ramanujan’s theta-function identities. Particularly, we prove that, for any integer α ≥ 0, ¯p(9α+1(24n + 23)) ≡ 0 (mod 3) and ¯p(9α+1(24n + 22) + 1) ≡ 0 (mod 3). Furthermore, we prove some new congruences modulo 5 and 7 for ¯p(n). For example, we prove that ¯p(5n+k+3) ≡ 0 (mod 5), where k = 3n2 ± n.

References