On the vanishing of some mock theta functions at odd roots of unity

2021 ◽  
Vol 7 (3) ◽  
Author(s):  
Mohamed El Bachraoui
Keyword(s):  
Theta Functions ◽  
Roots Of Unity ◽  
Mock Theta Functions

scholarly journals Quantum q-series identities

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Jeremy Lovejoy
Keyword(s):  
Power Series ◽  
Unit Disk ◽  
Hypergeometric Series ◽  
Dense Subset ◽  
Theta Functions ◽  
Roots Of Unity ◽  
Mock Theta Functions ◽  
Classical Sense ◽  
False Theta Functions ◽  
Series Transformation

As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

Tenth Order Mock Theta Functions: Part IV

2018 ◽  
pp. 229-248
Author(s):  
George E. Andrews ◽  
Bruce C. Berndt
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions

scholarly journals CERTAIN RELATIONS FOR MOCK THETA FUNCTIONS OF ORDER EIGHT

2009 ◽  
Vol 24 (4) ◽  
pp. 629-640
Author(s):  
Maheshwar Pathak ◽  
Pankaj Srivastava
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions

scholarly journals Congruences for sixth order mock theta functions $ \lambda(q) $ and $ \rho(q) $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Harman Kaur ◽  
Meenakshi Rana
Keyword(s):  
Theta Functions ◽  
Sixth Order ◽  
Mock Theta Functions ◽  
Lost Notebook

<p style='text-indent:20px;'>Ramanujan introduced sixth order mock theta functions <inline-formula><tex-math id="M3">\begin{document}$ \lambda(q) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \rho(q) $\end{document}</tex-math></inline-formula> defined as:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \lambda(q) &amp; = \sum\limits_{n = 0}^{\infty}\frac{(-1)^n q^n (q;q^2)_n}{(-q;q)_n},\\ \rho(q) &amp; = \sum\limits_{n = 0}^{\infty}\frac{ q^{n(n+1)/2} (-q;q)_n}{(q;q^2)_{n+1}}, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>listed in the Lost Notebook. In this paper, we present some Ramanujan-like congruences and also find their infinite families modulo 12 for the coefficients of mock theta functions mentioned above.</p>

scholarly journals Representations of affine superalgebras and mock theta functions II

2016 ◽  
Vol 300 ◽  
pp. 17-70 ◽  
Author(s):  
Victor G. Kac ◽  
Minoru Wakimoto
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions

Asymptotic Transformations of q-Series

1998 ◽  
Vol 50 (2) ◽  
pp. 412-425 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Asymptotic Expansions ◽  
General Class ◽  
Theta Functions ◽  
Mock Theta Functions

AbstractFor the q–series we construct a companion q–series such that the asymptotic expansions of their logarithms as q → 1– differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new q–hypergeometric identity. We give an asymptotic expansion of a general class of q–series containing some of Ramanujan's mock theta functions and Selberg's identities.

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