We define odd-balanced unimodal sequences and show that their generating function V ( x , q ) \mathcal {V}(x,q) has the same remarkable features as the generating function for strongly unimodal sequences U ( x , q ) U(x,q) . In particular, we discuss (mixed) mock modularity, quantum modularity, and congruences modulo 2 2 and 4 4 . We also study two related functions which share some of the properties of U ( x , q ) U(x,q) and V ( x , q ) \mathcal {V}(x,q) .
In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples f (q), Ramanujan claimed that there is a collection {G j } of modular forms such that for each root of unity ζ, there is a j such that limMoreover, Ramanujan claimed that this collection must have size larger than 1. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection cannot have size 1. These results arise within the context of Zagier's theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for "Ramanujan's radial limits" as L-values.
This paper contains three main results: the first one is to derive two "period relations" and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of "Jacobi integrals" on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with "period relations" is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607-632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the "Mordell integrals" associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et al. in Commun Math Phys 255(2):469-512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.
Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on SL 2 (Z) with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called Zagier duality. After the result of Zagier, this type of duality was studied broadly in various viewpoints, including the theory of a mock modular form. In this paper, we consider this problem with Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using the holomorphic Poincaré series and its supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights −k and k + 2, respectively, k > 0, for an H-group. We also investigate the structures of them such as the images under the differential operators D k+1 and ξ −k and quadric relations of the critical values of their L-functions.
Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, $H$-groups by employing the theory of supplementary functions introduced and developed by M. I. Knopp and S. Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms
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