Some Smallest Parts Functions from Variations of Bailey’s Lemma

Abstract

We construct new smallest parts partition functions and smallest parts crank functions by considering variations of Bailey's Lemma and conjugate Bailey pairs. The functions we introduce satisfy simple linear congruences modulo 3 and 5. We introduce and give identities for two four variable q-hypergeometric functions; these functions specialize to some of our new spt-crank-type functions as well as many known spt-crank-type functions.

On a modularity conjecture of Andrews, Dixit, Schultz, and Yee for a variation of Ramanujan's ω(q)

“…The generating function in (1.5) previously appeared in [16], where several additional families of spt-functions arising from Bailey pairs were studied systematically. Indeed, one of the functions defined in [16] is…”

Ranks For Two Partition Quadruple Functions

“…where ζ ℓ is a primitive ℓ th root of unity, as rank functions to prove the congruences u(3n) ≡ 0 (mod 3), u(5n) ≡ u(5n+3) ≡ 0 (mod 5), and u(7n) ≡ u(7n+5) ≡ 0 (mod 7) respectively. These functions correspond to the z = ζ 3 , ζ 5 , and ζ 7 cases of F (z 2 , z −2 , z; q), where F (ρ 1 , ρ 2 , z; q) is a function the author studied in [8] defined by…”