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

Volume 173 / 2016

Chris Jennings-Shaffer Acta Arithmetica 173 (2016), 317-364 MSC: Primary 11P81, 11P83; Secondary 33D15. DOI: 10.4064/aa8193-2-2016 Published online: 8 June 2016

Abstract

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.

Authors

  • Chris Jennings-ShafferDepartment of Mathematics
    Oregon State University
    Kidder Hall 368
    Corvallis, OR 97331, U.S.A.
    e-mail

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