scholarly journals A general double sum identity, mock theta functions, and Bailey pairs

2021 ◽  
Vol 198 (4) ◽  
pp. 377-385
Author(s):  
Alexander E. Patkowski
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Bailey Pairs ◽  
Double Sum

scholarly journals MOCK THETA DOUBLE SUMS

2016 ◽  
Vol 59 (2) ◽  
pp. 323-348 ◽  
Author(s):  
JEREMY LOVEJOY ◽  
ROBERT OSBURN
Keyword(s):  
General Result ◽  
Theta Functions ◽  
Similar Type ◽  
Mock Theta Functions ◽  
Bailey Pairs ◽  
Special Cases ◽  
Seventh Order

AbstractWe prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

scholarly journals Asymptotics for Bailey-type mock theta functions

2021 ◽  
Author(s):  
Taylor Garnowski
Keyword(s):  
Asymptotic Expansions ◽  
Fourier Coefficients ◽  
Circle Method ◽  
Theta Functions ◽  
Auxiliary Function ◽  
Asymptotic Estimates ◽  
Mock Theta Functions ◽  
Bailey Pairs ◽  
Jacobi Theta Functions ◽  
Do So

AbstractWe compute asymptotic estimates for the Fourier coefficients of two mock theta functions, 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 encounter cancelation in our estimates for one of the mock theta functions due to the auxiliary function $$\theta _{n,p}$$ θ n , p arising from the splitting of Hickerson and Mortenson. We deal with this by using higher-order asymptotic expansions for the Jacobi 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
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