scholarly journals Sixth order mock theta functions

2007 ◽  
Vol 216 (2) ◽  
pp. 771-786 ◽  
Author(s):  
Bruce C. Berndt ◽  
Song Heng Chan
Keyword(s):  
Theta Functions ◽  
Sixth Order ◽  
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>

ON SOME NEW MOCK THETA FUNCTIONS

2018 ◽  
Vol 107 (1) ◽  
pp. 53-66
Author(s):  
NANCY S. S. GU ◽  
LI-JUN HAO
Keyword(s):  
Theta Functions ◽  
Constant Term ◽  
Second Order ◽  
Sixth Order ◽  
Mock Theta Functions ◽  
Dual Nature ◽  
Partial Theta Functions

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

scholarly journals Combinatorics of some fifth and sixth order mock theta functions

2021 ◽  
Vol 29 (1) ◽  
pp. 1803-1818
Author(s):  
Meenakshi Rana ◽  
◽  
Shruti Sharma ◽  
Keyword(s):  
Theta Functions ◽  
Sixth Order ◽  
Mock Theta Functions

scholarly journals Sixth Order Mock Theta Functions

2018 ◽  
pp. 109-147
Author(s):  
George E. Andrews ◽  
Bruce C. Berndt
Keyword(s):  
Theta Functions ◽  
Sixth Order ◽  
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.

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