Related papers: Bilateral series and Ramanujan's radial limits
Ramanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. For example, we find the four tenth-order mock theta functions and their six identities. The six…
Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. For example, we find the four tenth-order mock theta…
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results for…
In this note we consider infinite series similar to the "strange" function $F(q)$ of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-Rhoades, Rolen-Schneider, and others in connection to quantum modular forms. We…
We present completions of mock theta functions to harmonic weak Maass forms of weight $1/2$ and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $1/2$ that have mock theta…
We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion…
Using properties of Appell-Lerch functions, we give insightful proofs for six of Ramanujan's identities for the tenth-order mock theta functions.
Recently, MacMahon's generalized sum-of-divisor functions were shown to link partitions, quasimodular forms, and q-multiple zeta values. In this paper, we explore many further properties and extensions of these. Firstly, we address a…
Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to…
Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of…
Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson,…
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition…
In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class…
Recently Gordon and McIntosh introduced the third order mock theta function $\xi(q)$ defined by $$ \xi(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of…
The arithmetic properties of the second order mock theta function $\mathcal{B}(q)$, introduced by McIntosh, defined by \begin{equation*} \mathcal{B}(q) := \sum_{n \geq 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum_{n \geq 0}b(n)q^n,…
We observe that certain famous evaluations of the Rogers-Ramanujan continued fraction $R(q)$ are close to $2\pi-6$ and $(2\pi-6)/2\pi$, and that $2\pi-6$ can be expressed by a Rogers-Ramanujan continued fraction in which $q$ is very nearly…
In this work we study the Plancherel-Rotach type asymptotics for selected $q$-series and $q$-orthogonal polynomials with complex scalings. The $q$-series we cover are Euler's $q$-exponential, Ramanujan function, Jackson's $q$-Bessel…
In this paper we consider the first four of the eight identities between the tenth order mock theta functions, found in Ramanujan's lost notebook. These were originally proved by Choi. Here we give an alternative (much shorter) proof.
We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions.…
We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two…