Integration of modular forms with theta series

Dirichlet series, having holomorphic analytic continuation to the whole complex plane and satisfying a functional equation of standard type, are obtained by considering Rankin type integrals of products of elliptic modular forms for the group SL₂()by theta series of integral quadratic forms of determinant ±1. In a series of cases the constructed Dirichlet series are Mellin transforms of elliptic modular forms of higher weight than the initial forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 5–21, 1990.     

Periods of modular forms and Jacobi theta functions

Oblatum 20-III-1990 & 23-VII-1990     

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.     

Sixth order mock theta functions

Two new mock theta functions of the sixth order are defined. The main theorem in this paper (Theorem 1.1) provides four transformation formulas relating the new mock theta functions with Ramanujan's mock theta functions of the sixth order. Two further representations of the new mock theta functions are established. Lastly, a hitherto unproved entry from Ramanujan's lost notebook related to sixth order mock theta functions is proved.     

Secord-order cusp forms and mixed mock modular forms

In this paper, we consider the space of second order cusp forms. We determine that this space is precisely the same as a certain subspace of mixed mock modular forms. Based upon Poincaré series of Diamantis and O’Sullivan (Trans. Am. Math. Soc. 360:5629–5666, 2008) which span the space of second order cusp forms, we construct Poincaré series which span a natural (more general) subspace of mixed mock modular forms.     

Asymptotics for Bailey-type mock theta functions

The Ramanujan Journal - We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the...     

The Bailey chain and mock theta functions

Standard applications of the Bailey chain preserve mixed mock modularity but not mock modularity. After illustrating this with some examples, we show how to use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to explicitly construct families of q$q$-hypergeometric multisums which are mock theta functions. We also prove identities involving some of these multisums and certain classical mock theta functions.     

Eulerian series as modular forms

In 1988, Hickerson proved the celebrated ``mock theta conjectures' in a collection of ten identities from Ramanujan's ``lost notebook' which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.

     

On certain Ramanujan's mock theta functions

Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. Recently, Andrews and Hickerson found a set of seven more functions in Ramanujan's Lost Note Book and formally labelled them as mock theta functions of order six. In this paper the complete forms of these functions have been studied and connected with the bilateral basic hypergeometric series₂Ψ₂. Several other interesting properties and transformations have also been studied.     

Supersingular elliptic curves, theta series and weight two modular forms

Let be a prime, and let denote the space of weight two modular forms on all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that is spanned as a -module by theta series attached to the unique quaternion algebra that is ramified at , at infinity, and at no other primes.

     

Combinatorics of tenth-order mock theta functions

In this paper, we provide the combinatorial interpretations of two tenth-order mock theta functions which appeared in some identities given in Ramanujan’s lost notebook ((1988) Narosa Publishing House, New Delhi).     

Unclosed asymptotic expansions and mock theta functions

In his last letter to Hardy, Ramanujan defined 17 functions f(q), (|q|<1), which he called mock theta functions. Each f(q) has infinitely many exponential singularities at roots of unity, and under radial approach to every such singularity, f(q) has an asymptotic approximation consisting of a finite number of terms with closed exponential factors, plus an error term O(1). We give an example of a q-series in Eulerian form having an approximation with an unclosed exponential factor. Complete asymptotic expansions as q→1 of some shifted q-factorials are given in terms of polylogarithms and Bernoulli polynomials. Supported by the Natural Sciences and Engineering Research Council of Canada.     

Combinatorial interpretations as two-line array for the mock theta functions

The mock theta functions introduced by Ramanujan have been studied by many authors both analytically and combinatorially. The combinatorial interpretations that are known for some of them are quite different in nature. In this paper we present combinatorial interpretation as two-line array for many of the classical mock theta functions.     

Hilbert modular forms of weight 1/2 and theta functions

Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result—under certain mild restrictions on the level and character—to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.     

On two fifth order mock theta functions

We consider the fifth order mock theta functions χ ₀ and χ ₁, defined by Ramanujan, and find identities for these functions, which relate them to indefinite theta functions. Similar identities have been found by Andrews for the other fifth order mock theta functions and the seventh order functions.     

On second and eighth order mock theta functions

The Ramanujan Journal - 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...     

Construction of vector-valued modular integrals and vector-valued mock modular forms

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms.