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Ramanujan-type congruences for overpartitions modulo 16

机译：Ramanujan-type congruences for overpartitions modulo 16

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Let (p) over bar (n) denote the number of overpartitions of n. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for (p) over bar (n) and derived a number of congruences for (p) over bar (n) modulo 4, 8 and 64 including (p) over bar (8n + 7) equivalent to 0 (mod 64) for n >= 0. In this paper, we give a 16-dissection of the generating function for (p) over bar (n) modulo 16 and showthat (p) over bar (16n + 14) equivalent to 0 (mod 16) for n >= 0. Moreover, using the 2-adic expansion of the generating function for (p) over bar (n) according to Mahlburg, we obtain that (p) over bar (l(2)n + rl) = 0 (mod 16), where n >= 0, l equivalent to -1 (mod 8) is an odd prime and r is a positive integer with l r. In particular, for l = 7 and n >= 0, we get (p) over bar (49n + 7) = 0 (mod 16) and (p) over bar (49n + 14) equivalent to 0 (mod 16). We also find four congruence relations: (p) over bar (4n) equivalent to (-1)(n) (p) over bar (n) (mod 16) for n >= 0, (p) over bar (4n) equivalent to (-1)(n) (p) over bar (n) (mod 32) where n is not a square of an odd positive integer, (p) over bar (4n) = (-1)(n) (p) over bar (n) (mod 64) for n not equivalent to 1, 2, 5 (mod 8) and (p) over bar (4n) equivalent to (-1)(n) (p) over bar (4n) equivalent to (mod 128) for n equivalent to 0 (mod 4).

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