scholarly journals A Mock Theta Function of Second Order

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Bhaskar Srivastava
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Theta Series ◽  
Mathematical Physics ◽  
Second Order ◽  
Mock Theta Function ◽  
Third Order ◽  
Quantum Invariant ◽  
Order Mock Theta Function

We consider the second-order mock theta function (), which Hikami came across in his work on mathematical physics and quantum invariant of three manifold. We give their bilateral form, and show that it is the same as bilateral third-order mock theta function of Ramanujan. We also show that the mock theta function () outside the unit circle is a theta function and also write as a coefficient of of a theta series. First writing as a coefficient of a theta function, we prove an identity for .

Combinatorics of two second-order mock theta functions

2020 ◽  
pp. 1-11
Author(s):  
Hannah Burson
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Second Order ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Third Order ◽  
Combinatorial Interpretations ◽  
Special Cases ◽  
Order Mock Theta Function

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function [Formula: see text] and Ramanujan’s third-order mock theta function [Formula: see text].

ARITHMETIC PROPERTIES OF COEFFICIENTS OF THE MOCK THETA FUNCTION

2019 ◽  
Vol 102 (1) ◽  
pp. 50-58
Author(s):  
RENRONG MAO
Keyword(s):  
Theta Function ◽  
Second Order ◽  
Arithmetic Progressions ◽  
Mock Theta Function ◽  
Arithmetic Properties ◽  
Order Mock Theta Function

We investigate the arithmetic properties of the second-order mock theta function $B(q)$ and establish two identities for the coefficients of this function along arithmetic progressions. As applications, we prove several congruences for these coefficients.

scholarly journals On second order mock theta function $ B(q) $

2021 ◽  
Vol 30 (1) ◽  
pp. 52-65
Author(s):  
Harman Kaur ◽  
◽  
Meenakshi Rana
Keyword(s):  
Theta Function ◽  
Second Order ◽  
Mock Theta Function ◽  
Arithmetic Properties ◽  
Order Mock Theta Function

<abstract><p>In this paper, we present some arithmetic properties for the second order mock theta function $ B(q) $ given by McIntosh as:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ B(q) = \sum\limits_{n = 0}^{\infty}\frac{q^n(-q;q^2)_n}{(q;q^2)_{n+1}}. $\end{document} </tex-math></disp-formula></p> </abstract>

scholarly journals Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket

2018 ◽  
Vol 14 (07) ◽  
pp. 1961-1981
Author(s):  
Robert Schneider
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Modular Forms ◽  
Statistical Physics ◽  
Theta Functions ◽  
Triple Product ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Quantum Modular Forms ◽  
Unimodal Sequences

In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function [Formula: see text] of Gordon–McIntosh. Here we show that [Formula: see text] arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the [Formula: see text]-bracket of Bloch–Okounkov. Second, we find [Formula: see text] to extend in [Formula: see text] to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other [Formula: see text]-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.

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