Fifth Order Mock Theta Functions: Proof of the Mock Theta Conjectures

2018 ◽  
pp. 77-108
Author(s):  
George E. Andrews ◽  
Bruce C. Berndt
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Fifth Order

On two fifth order mock theta functions

2009 ◽  
Vol 20 (2) ◽  
pp. 207-214 ◽  
Author(s):  
Sander Zwegers
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Fifth Order

Bilateral series and Ramanujan radial limits of mock (false) theta functions

2019 ◽  
Vol 30 (04) ◽  
pp. 1950023
Author(s):  
Bin Chen
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Radial Limits ◽  
The Third ◽  
False Theta Functions ◽  
Even Order ◽  
Fifth Order ◽  
Vector Valued

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

Partitions associated with two fifth-order mock theta functions and Beck-type identities

2019 ◽  
Vol 16 (04) ◽  
pp. 841-855
Author(s):  
Runqiao Li ◽  
Andrew Y. Z. Wang
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Partition Identity ◽  
Fifth Order

In this paper, we first generalize Ramanujan’s partition identity derived from the fifth order mock theta functions [Formula: see text] and [Formula: see text]. Then we establish Beck-type identities based on our general partition identity. All the results are proved both analytically and combinatorially.

scholarly journals Applications of Generalized q-Difference Equations for General q-Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1222
Author(s):  
Zeya Jia ◽  
Bilal Khan ◽  
Qiuxia Hu ◽  
Dawei Niu
Keyword(s):  
Generating Function ◽  
Difference Equations ◽  
Generating Functions ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Fifth Order

Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(∞,−1,1,q) is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation.

Fifth Order Mock Theta Functions: Elementary Identities

2018 ◽  
pp. 17-33
Author(s):  
George E. Andrews ◽  
Bruce C. Berndt
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Fifth Order

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
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