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Exotic Bailey-Slater spt-functions III: Bailey pairs from groups B, F, G, and J

100%

Jennings-Shaffer C.

Acta Arithmetica

2016

tom 173

nr 4

317-364

We continue to investigate spt-type functions that arise from Bailey pairs. In this third paper on the subject, we proceed to introduce additional spt-type functions. We prove simple Ramanujan type congruences for these functions which can be explained by an spt-crank-type function. The spt-crank-type functions are actually defined first, with the spt-type functions coming from setting z = 1 in this definition. We find some of the spt-crank-type functions to have interesting representations as single series, some of which reduce to infinite products. Additionally, we find dissections of the other spt-crank-type functions when z is a certain root of unity. Both methods are used to explain congruences for the spt-type functions. Our series formulas require Bailey's Lemma and conjugate Bailey pairs. Our dissection formulas follow from Bailey's Lemma and dissections of known ranks and cranks.

The spt-crank for overpartitions

63%

Garvan F., Jennings-Shaffer C.

Acta Arithmetica

2014

tom 166

nr 2

141-188

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\overline{spt}(n)$, $\overline{spt}_1(n)$, $\overline{spt}_{2}(n)$, and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions. In this paper we define spt-cranks in terms of vector partitions that combinatorially explain the known simple congruences for all the spt-overpartition functions as well as new simple congruences. For all the overpartition functions except M2spt(n) we are able to define the spt-crank purely in terms of marked overpartitions. The proofs of the congruences depend on Bailey's Lemma and the difference formulas for the Dyson rank of an overpartition (Lovejoy and Osburn, 2008) and the $M_2$-rank of a partition without repeated odd parts (Lovejoy and Osburn, 2009).

Ograniczanie wyników

2 Acta Arithmetica

2 Jennings-Shaffer C.

1 Garvan F.

1 2016

1 2014

Wyniki wyszukiwania

Exotic Bailey-Slater spt-functions III: Bailey pairs from groups B, F, G, and J

The spt-crank for overpartitions