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

Chris Jennings-Shaffer¹

¹ Department of Mathematics Oregon State University Kidder Hall 368 Corvallis, OR 97331, U.S.A.

Acta Arithmetica, Tome 173 (2016) no. 4, pp. 317-364.

Résumé

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.

Détail

DOI : 10.4064/aa8193-2-2016

Keywords: continue investigate spt type functions arise bailey pairs third paper subject proceed introduce additional spt type functions prove simple ramanujan type congruences these functions which explained spt crank type function spt crank type functions actually defined first spt type functions coming setting definition spt crank type functions have interesting representations single series which reduce infinite products additionally dissections other spt crank type functions certain root unity methods explain congruences spt type functions series formulas require bailey lemma conjugate bailey pairs dissection formulas follow bailey lemma dissections known ranks cranks