ON (2,5)-REGULAR BIPARTITIONS WITH ODD PARTS DISTINCT
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In his work, K. Alladi studied the partition function pod(n), the number of partitions of an integer n with odd parts distinct (the even parts are unrestricted). He obtained a series expansion for the product generating function of partitions in which the odd parts do not repeat. Later, Hirschhorn and Sellers obtained some internal congruences involving the infinite families and Ramanujan-type congruences for pod(n). Let B(n) denote the number of (2, 5)-regular bipartitions of a positive integer n with odd parts distinct (even parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for B(n). For example, for modulo 16, ∞∑n=0 B(16·34α·52β·72ϒn+6·34α·52β·72ϒ+1)qn≡8f91, where α, β, ϒ≥0.
Abstract
In his work, K. Alladi studied the partition function pod(n), the number of partitions of an integer n with odd parts distinct (the even parts are unrestricted). He obtained a series expansion for the product generating function of partitions in which the odd parts do not repeat. Later, Hirschhorn and Sellers obtained some internal congruences involving the infinite families and Ramanujan-type congruences for pod(n). Let B(n) denote the number of (2, 5)-regular bipartitions of a positive integer n with odd parts distinct (even parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for B(n). For example, for modulo 16, ∞∑n=0 B(16·34α·52β·72ϒn+6·34α·52β·72ϒ+1)qn≡8f91, where α, β, ϒ≥0.
참고문헌 (14)
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