The Bailey transform and Hecke–Rogers identities for the universal mock theta functions

Abstract

Abstract. Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions g 2 (z; q), g 3 (z; q), K(z; q) by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these identities by using Bailey pair technology. In this paper, we give proofs of Garvan's identities by applying Bailey's transform with the conjugate Bailey pair of Warnaar and three Bailey pairs deduced from two special cases of 6 ψ 6 given by Slater. In particular, we obtain a compact form of twovariable Hecke-Rogers identity related to g 3 (z; q), which imply the corresponding identity given by Garvan. We also extend these two-variable Hecke-Rogers identities into infinite families.

On the Andrews–Warnaar identities for partial theta functions

“…Second, in [20,23] Kim and Lovejoy extended Warnaar's results in [40] to those in terms of partial and ternary indefinite theta functions. Third, Ji and Zhao [16] used the Bailey transform and the conjugate Bailey pair of Warnaar to show Garvan's two-variable Hecke-Rogers identities for the universal mock theta functions (also see [13,15]). These works convince us that Kim-Lovejoy's and Warnaar's methods pave a new way to Bailey pairs and Bailey chains.…”

Some New Transformation Formulas for <i>q</i>-Series through the Bailey Transform

“…Bailey [4] [10], Bressoud [2], and Slater [11] [12] used this transform to derive a number of identities of Rogers-Ramanujan type identities. Ji and Zhao [13] established the Hecke-Rogers identities for the universal mock theta functions by means of the Bailey transform.…”

Partial Indefinite Theta Identities

“…For a general theory of multi-variable indefinite theta functions, see [27]. We make use of an identity of Ji and Zhao [13]. Arguing as in Warnaar's proof of (1.2), they proved that if (α n , β n ) is a Bailey pair relative to q, then…”

“…In Section 5 we use a result of Ji and Zhao [13] to recast all of the partial indefinite theta identities from Sections 1-4 in terms of indefinite ternary theta functions. In doing so, we lose the first partial indefinite theta term on the right-hand side but gain considerably in simplicity.…”