On second and eighth order mock theta functions

Abstract

Mock theta functions have been deeply studied in the literature. Historically, there are many forms of representations for mock theta functions: Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, we first establish Hecke-type double sums for the second and eighth order mock theta functions by Bailey’s lemma and a Bailey pair given by Andrews and Hickerson. Meanwhile, we give different proofs of the generalized Lambert series for the mock theta functions A(q), $$U_0(q)$$ , and $$U_1(q)$$ . Furthermore, using Ramanujan’s $${_1}\psi _1$$ summation formula and a $${_2}\psi _2$$ transformation formula due to Bailey, we prove some identities related to these mock theta functions.