Transformation Formulas: 10th Order Mock Theta Functions

In 2004, the first author gave the combinatorial interpretations of four mock theta functions of Srinivasa Ramanujan using $n$-color partitions which were introduced by himself and G.E. Andrews in 1987. In this paper we introduce a new class of partitions and call them "split $(n+t)$-color partitions". These new partitions generalize Agarwal-Andrews $(n+t)$-color partitions. We use these new combinatorial objects and give combinatorial meaning to two basic functions of Gordon-McIntosh found in 2000. They used these functions to establish the modular transformation formulas for certain eight order mock theta functions. The work done here has a great potential for future research.

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Higher depth mock theta functions and q-hypergeometric series

Forum Mathematicum ◽

10.1515/forum-2021-0013 ◽

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

Research in Number Theory ◽

10.1007/s40993-021-00279-5 ◽

2021 ◽

Vol 7 (3) ◽

Author(s):

Mohamed El Bachraoui

Keyword(s):

Theta Functions ◽

Roots Of Unity ◽

Mock Theta Functions

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Tenth Order Mock Theta Functions: Part IV

Ramanujan's Lost Notebook ◽

10.1007/978-3-319-77834-1_11 ◽

2018 ◽

pp. 229-248

Author(s):

George E. Andrews ◽

Bruce C. Berndt

Keyword(s):

Theta Functions ◽

Mock Theta Functions

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CERTAIN RELATIONS FOR MOCK THETA FUNCTIONS OF ORDER EIGHT

Communications of the Korean Mathematical Society ◽

10.4134/ckms.2009.24.4.629 ◽

2009 ◽

Vol 24 (4) ◽

pp. 629-640

Author(s):

Maheshwar Pathak ◽

Pankaj Srivastava

Keyword(s):

Theta Functions ◽

Mock Theta Functions

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

Electronic Research Archive ◽

10.3934/era.2021084 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Harman Kaur ◽

Meenakshi Rana

Keyword(s):

Theta Functions ◽

Sixth Order ◽

Mock Theta Functions ◽

Lost Notebook

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) $\end{document}</tex-math></inline-formula> defined as:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \lambda(q) & = \sum\limits_{n = 0}^{\infty}\frac{(-1)^n q^n (q;q^2)_n}{(-q;q)_n},\\ \rho(q) & = \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>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.

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