Related papers: Mock Theta Function Identities Deriving from Bilat…
Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these…
The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an…
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…
We find the Hecke-Rogers type series representations of generating functions of the Hurwitz class numbers which is very close to certain mock theta functions. We also prove two combinatorial interpretation of Hurwitz class numbers appeared…
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric…
Mixed mock modular forms are functions which lie in the tensor space of mock modular forms and modular forms. As q-hypergeometric series, mixed mock modular forms appear to be much more common than mock theta functions. In this survey, we…
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,…
In the explicit formula for the signed mock theta functions $\Phi^{(-)[m,s]}$ obtained from the coroot lattice of $D(2,1;a)$, functions with indefinite quadratic forms naturally take place. We compute their modular transformation properties…
The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ %(where $(q,x)\in {\bf C}^2$, $|q|<1$) defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. For $q\in (-1,0)$ the…
In a private communication, K. Ono conjectured that any mock theta function of weight 1/2 or 3/2 can be congruent modulo a prime $p$ to a weakly holomorphic modular form for just a few values of $p$. In this paper we describe when such a…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
We consider the partial theta function $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. We show that, for any fixed $q$, if $\zeta$ is a multiple…
In the paper, we give partition-theoretic results for the coefficients of some mock theta functions and prove their congruence properties. Some recurrence relations connecting the coefficients of the mock theta functions with certain…
We compute asymptotic estimates for the Fourier coefficients of two mixed mock modular forms, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle method due to Wright and a modified Tauberian theorem.…
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula.…
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2\phi_1$…
We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic…
In this paper, transformation formulas for the function \[ A_{1}\left(z,s:\chi\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}\chi\left(n\right)\chi\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2\pi imnz/k} \] are obtained. Sums…
We obtain an explicit formula for the mock theta function $\Phi^{[m,s]}$ in the case when either $m$ or $s$ is a half of an odd integer by using the coroot lattice of $D(2,1;a)$. This enables us, together with the recurrence formula for…
This paper discuss a new class of functional equations by using both Poisson summation formula and Jacobi type theta a function. The class of Riemann type functional equations are derived from self-reciprocal probability density functions.…