Tenth order mock theta functions in Ramanujan’s lost notebook (IV)

We obtain four Hecke-type double sums for three of Ramanujan's third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews' work on $q$-orthogonal polynomials and Bringmann, Hikami, and Lovejoy's work on unified Witten-Reshetikhin-Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.

S. Ramanujan introduced the mock theta functions in his last letter to G.H. Hardy. In his letter and Lost Notebook, he presented mock theta functions and their identities. The expression of each mock theta function relies on the hypergeometric series. G.N. Watson studied the relations between Ramanujan’s fifth order mock theta functions and the function introduced by M. Lerch in 1893. In this manuscript, we develop the relations between the basic bilateral hypergeometric series and the functions deduced from all mock theta functions introduced by G.E. Andrews, B.C. Berndt, S.H. Chan, R. McIntosh, Ramanujan, and Watson.

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e2πir (r rational), there is a theta function Fr(q) with F(q) − Fr(q) = O(1). In this paper we establish the relationship between two families of mock theta functions.

On page 9 in his Lost Notebook [232], Ramanujan offers eight identities for tenth order mock theta functions.

In this paper we express two new mock theta functions with one parameter as the Appell-Lerch sums using the Bailey mechanism. Meanwhile, we also obtain some Hecke-type double sums for some new $q$-series. In addition, we establish the relationships between the mock theta functions and the classical sixth and eighth order mock theta functions.Furthermore, we give the Hecke-type double sums for the second order mock theta function $D_5(q)$.

Harmonic weak Maas forms have recently been shown to have quite a few interesting arithmetic applications, including their connection to Ramanujan’s work on (mock) theta functions (Bringmann and Ono in Invent. Math. 165(2):243–266, 2006; Bringmann and Ono in Ann. Math. 171(1):419–449, 2010; Bruinier and Ono in Heegner divisors, L-functions and harmonic weak Maass forms, Ann. Math., 2011, to appear; Zagier in Ramanujan’s mock theta functions and their applications [d’apres Zwegers and Bringmann–Ono] Seminaire Bourbaki 986, 2006–2007).

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

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.

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of \(\mathrm {SL}_2(\mathbb {Z})\), up to nontrivial error terms; however, their domains (the upper half-plane \(\mathbb {H}\) and the rationals \(\mathbb {Q}\), respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more.

Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan’s two-parameter mock theta functions are the special cases of these functions.

We consider the second-order mock theta function (), which Hikami came across in his work on mathematical physics and quantum invariant of three manifold. We give their bilateral form, and show that it is the same as bilateral third-order mock theta function of Ramanujan. We also show that the mock theta function () outside the unit circle is a theta function and also write as a coefficient of of a theta series. First writing as a coefficient of a theta function, we prove an identity for .

Short proofs of Ramanujan-like identities for the eighth order mock theta function $$V_{0}(q)$$

Congruences for the second order mock theta function B(q)

<abstract><p>Using the three-term Weierstrass relation for theta functions and the properties of Hecke-type double sums and Appell-Lerch sums, we give new and simple proofs for the seventh order mock theta conjectures.</p></abstract>