Richard J. McIntosh

A Survey of Classical Mock Theta Functions

In his last letter to Hardy, Ramanujan defined 17 functions M(q), | q | < 1, which he called mock θ-functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an exponential singularity, there is a θ-function T ζ(q) with \(M(q) - {T}_{\zeta }(q) = O(1)\). Since then, other functions have been found which possess this property. We list various linear relations between these functions and develop their transformation laws under the modular group. We show that each mock θ-function is related to a member of a universal family (mock θ-conjectures). In recent years the subject has received new impetus and importance through a strong connection with the theory of Maass forms. The final section of this survey provides some brief remarks concerning these new developments.

Modular Transformations of Ramanujan’s Fifth and Seventh Order Mock Theta Functions

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 obtain the transformations of Ramanujans fifth and seventh order mock theta functions under the modular group generators τ → τ + 1 and τ → −1/τ, where q = eπiτ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.

Some Asymptotic Formulae for q-Shifted Factorials

The q-shifted factorial defined by (a : qk)n = (1 − a) (1 − aqk)(1 − aq2k)... (1 − aq(n − 1)k) appears in the terms of basic hypergeometric series. Complete asymptotic expansions as q → 1 of some q-shifted factorials are given in terms of polylogarithms and Bernoulli polynomials.

Algebraic Dilogarithm Identities

AbstractThe Rogers L-function

On the Universal Mock Theta Function \(g_{{{\scriptstyle 2}}}\) and Zwegers’ \(\mu \)-Function

L(x) = \sum\limits_{n = 1}^\infty {\frac{{x^n }} {{n^2 }} + \frac{1} {2}\log x} \log (1 - x)

Some Eighth Order Mock Theta Functions

satisfies the functional equation

A generalization of a congruential property of Lucas

L(x) + L(y) = L(xy) + L\left( {\frac{{x(1 - y)}} {{1 - xy}}} \right) + L\left( {\frac{{y(1 - x)}} {{1 - xy}}} \right)