American Journal of Mathematics and Statistics

p-ISSN: 2162-948X    e-ISSN: 2162-8475

2013;  3(3): 143-152

doi:10.5923/j.ajms.20130303.07

Relations Connecting Eighth Order Mock Theta Functions

Roselin Antony

Department of Mathematics, College of Natural and Computational Sciences, Mekelle University, Mekelle, P.O. Box 231, Ethiopia

Correspondence to: Roselin Antony, Department of Mathematics, College of Natural and Computational Sciences, Mekelle University, Mekelle, P.O. Box 231, Ethiopia.

Email:

Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.

Abstract

During the last years of his life, Ramanujan defined 17 functions F(q), where | q | < 1, and he named them as mock theta functions. The first detailed description of mock theta functions was given by Watson. In this paper, we obtain relations connecting mock theta functions, partial mock theta functions of order 8 and infinite products analogous to the identities of Ramanujan.

Keywords: Mock Theta Functions, Partial Mock Theta Functions, Infinite Products

Cite this paper: Roselin Antony, Relations Connecting Eighth Order Mock Theta Functions, American Journal of Mathematics and Statistics, Vol. 3 No. 3, 2013, pp. 143-152. doi: 10.5923/j.ajms.20130303.07.

Article Outline

1. Introduction
2. Methodology
3. Main Results and Discussion
4. Conclusions

1. Introduction

Ramanujan gave a list of seventeen mock theta functions and labeled them as third, fifth and seventh orders without giving any reason for his classification [1, 2]. Ramanujan’s general definition of a mock theta function is a function of defined by a q-series convergent when | q | < 1 which satisfies the following two conditions,
(a) For every root of unity, there exist a -function (q) such that difference between and (q) is bounded as q , radially.
(b) There is no single theta function which works for all , i.e. for every -function (q) there is some root of unity for which minus the theta function (q) is unbounded as q radially.
A study of these sums and expansions has been made by Watson[3], Agarwal[4] and Andrews[5]. Later on, Andrews and Hickerson[6], Choi[7] and Gordon and Mc Intosh[8] studied certain q-series in the Lost Notebook and named them as sixth, eighth and tenth order mock theta functions. Although Gordon and Mc Intosh[8] have given definitions of order of mock theta functions, and later Bringmann and Ono[9, 10] have given clarification for the order of the mock theta functions.
Also, relations connecting mock theta functions and partial mock theta functions are given by Srivastava[11] and Denis et al.[12]. Bhaskar Srivastava[13] provided relations connecting mock theta functions and partial mock theta functions of order 3, 5, 6 and 10 and relations connecting mock theta functions, partial mock theta functions of order 2, 3 and 6 and Ramanujan’s function μ(q). Recently, Roselin Antony and Atakalti Araya[14] obtained relations connecting mock theta functions of order 2 and infinite products analogous to the identities of Ramanujan. Also, Roselin Antony and Hailemariam Fiseha[15] obtained relations connecting mock theta functions of order 10 and infinite products analogous to the identities of Ramanujan. Also, Roselin Antony[16] obtained relations connecting mock theta functions of order 6 and infinite products analogous to the identities of Ramanujan
(1.1)
is a mock theta function, then the corresponding partial mock theta function is denoted by the terminating series,
(1.2)

2. Methodology

We shall make use of mock theta functions of order 8, their partial sums and infinite products in the known identity of Srivastava[11] to obtain new relations connecting mock theta functions of order 8.
Mock theta functions of order 8:
Gordon and McIntosh[8] found the following eight mock theta functions of order 8;
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
Ramanujan, in chapter 16 of his second notebook defined theta functions as follows; [17, 18]
(1.11)
An identity due to Euler is, [19, 20]
(1.12)
The special cases of the above identity are
(1.13)
(1.14)
The Famous Roger’s –Ramanujan identity is, [21, 22]
(1.15)
(1.16)
Hahn defined the septic analogue of the Rogers- Ramanujan functions as, [23, 24]
(1.17)
(1.18)
(1.19)
The Jackson – Slater identity;
Jackson[25] discovered the following identity;
(1.20)
This identity was independently rediscovered by Slater[26, Eqn.39] who also discovered its companion identity[Slater [26, Eqn.38]]
(1.21)
The identity analogous to the Rogers- Ramanujan identity is the so-called Gollnitz – Gordon identity given by, [27, 28]
(1.22)
(1.23)
The nonic analogue of Rogers – Ramanujan functions is
(1.24)
(1.25)
(1.26)
These equalities are due to Bailey [Bailey,[29]; Eqn.(1.6),(1.7) and (1.8)].
We shall make use of the following known identity of Srivastava[11];
(1.27)

3. Main Results and Discussion

We shall establish relations connecting mock theta functions, partial mock theta functions of order 8 and infinite products analogous to the identities of Ramanujan.
A) Taking in (1.27) and by (1.11), we get
(2.1)
i) Taking in (2.1)
and making use of (1.3), we get
(2.2)
ii) Taking in (2.1)
and making use of (1.4), we get
(2.3)
iii) Taking in (2.1)
and making use of (1.5), we get
(2.4)
iv) Taking in (2.1)
and making use of (1.6), we get
(2.5)
v) Taking in (2.1)
and making use of (1.7), we get
(2.6)
vi) Taking in (2.1)
and making use of (1.8), we get
(2.7)
vii) Taking in (2.1)
and making use of (1.9), we get
(2.8)
viii) Taking in (2.1)
and making use of (1.10), we get
(2.9)
B) Taking in (1.27)
and by (1.13), we get
(2.10)
i) Taking in (2.10)
and making use of (1.3), we get
(2.11)
ii) Taking in (2.10)
and making use of (1.4), we get
(2.12)
iii) Taking in (2.10)
and making use of (1.5), we get
(2.13)
iv) Taking in (2.10)
and making use of (1.6), we get
(2.14)
v) Taking in (2.10)
and making use of (1.7), we get
(2.15)
vi) Taking in (2.10)
and making use of (1.8), we get
(2.16)
vii) Taking in (2.10)
and making use of (1.9), we get
(2.17)
viii) Taking in (2.10)
and making use of (1.10), we get
(2.18)
Similarly, by assuming , we can establish relations connecting mock theta functions of order 6 and the infinite product C(q).
C) Taking in (1.27)
and by (1.15), we get
(2.19)
i) Taking in (2.19)
and making use of (1.3), we get
(2.20)
ii) Taking in (2.19)
and making use of (1.4), we get
(2.21)
iii) Taking in (2.19)
and making use of (1.5), we get
(2.22)
iv) Taking in (2.19)
and making use of (1.6), we get
(2.23)
v) Taking in (2.19)
and making use of (1.7), we get
(2.24)
vi) Taking in (2.19)
and making use of (1.8), we get
(2.25)
vii) Taking in (2.19)
and making use of (1.9), we get
(2.26)
viii) Taking in (2.19)
and making use of (1.10), we get
(2.27)
By taking , relations can be developed connecting mock theta functions and the infinite product E(q).
D) Taking in (1.27)
and by (1.17), we get
(2.28)
i) Taking in (2.28)
and by (1.3), we get
(2.29)
ii) Taking in (2.28)
and by (1.4), we get
(2.30)
iii) Taking in (2.28)
and by (1.5), we get
(2.31)
iv) Taking in (2.28)
and by (1.6), we get
(2.32)
v) Taking in (2.28)
and by (1.7), we get
(2.33)
vi) Taking in (2.28)
and by (1.8), we get
(2.34)
vii) Taking in (2.28)
and by (1.9), we get
(2.35)
viii) Taking in (2.28)
and by (1.10), we get
(2.36)
In the similar way, by assuming
and
, relations connecting mock theta functions of order six and the infinite products G(q) and H(q) can be obtained.
E) Taking in (1.27)
and by (1.20), we get
(2.37)
i) Taking in (2.37)
and by (1.3), we get
(2.38)
ii) Taking in (2.37)
and by (1.4), we get
(2.39)
iii) Taking in (2.37)
and by (1.5), we get
(2.40)
iv) Taking in (2.37)
and by (1.6), we get
(2.41)
v) Taking in (2.37)
and by (1.7), we get
(2.42)
vi) Taking in (2.37)
and by (1.8), we get
(2.43)
vii) Taking in (2.37)
and by (1.9), we get
(2.44)
viii) Taking in (2.37)
and by (1.10), we get
(2.45)
In the similar way, by taking
and , relations connecting mock theta functions of order six and the infinite products J(q), K(q) and L(q) can be obtained.
F) Taking in (1.27)
and by (1.24), we get
(2.46)
i) Taking in (2.46)
and by (1.3), we get
(2.47)
ii) Taking in (2.46)
and by (1.4), we get
(2.48)
iii) Taking in (2.46)
and by (1.5), we get
(2.49)
iv) Taking in (2.46)
and by (1.6), we get
(2.50)
v) Taking in (2.46)
and by (1.7), we get
(2.51)
vi) Taking in (2.46)
and by (1.8), we get
(2.52)
vii) Taking in (2.46)
and by (1.9), we get
(2.53)
viii) Taking in (2.46)
and by (1.10), we get
(2.54)
By assuming and , relations connecting mock theta functions of order six and the infinite products N(q) and P(q) can be obtained.
Similar works have been done on mock theta functions of order 8 by various mathematicians. Ahmad Ali[30] obtained relations connecting mock theta functions of order 8 and various mock theta functions of other orders. Maheshwar Pathak and Pankaj Srivastava[31] established relations connecting connecting partial mock theta functions and mock theta functions of order two, six, eight and ten.

4. Conclusions

In the similar way, many relations can be obtained using mock theta functions of different orders and infinite products analogous to Ramanujan’s identities.

References

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