two-variable generalization of new mock theta...

Tamsui Oxford Journal of Information and Mathematical Sciences 29(3) (2013) 389-402

Aletheia University

Two-Variable Generalization of New Mock

Theta Functions*

Bhaskar Srivastava†

Department of Mathematics and Astronomy

Lucknow University Lucknow, India

Received June 15, 2013, Accepted July 26, 2013

Dedicated to Professor H.M. Srivastava

Abstract

We define two variable generalization of new mock theta functions

recently developed by Andrews and also of two new mock theta functions

developed by Bringmann et al and give functional equations satisfied by

these generalized functions. We show that these generalized functions are

mock theta functions. We prove the main theorem that they are bounded

near the neighbourhood of the unit circle.

Keywords and Phrases: Mock theta functions, q- Hypergeometric series, Functional

equations.

* 2000 Mathematics Subject Classification. Primary 33D15.

† Email: [email protected]

390 Bhaskar Srivastava

1. Introduction

Ramanujan’s mathematics continues to generate a vast amount of research in a

variety of areas. Ramanujan made profound contributions but did not give rigorous

proofs. Undoubtedly the most famous are mock theta functions. In 1919, Ramanujan

returned to India, after about five years in England. In his last letter to Hardy, dated

January 12, 1920, he wrote: “I discovered very interesting functions recently which I

can call “Mock” theta-functions. Unlike the “False” theta-functions (studied partially

by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as

ordinary theta functions. I am sending you with this letter some examples”.

In the letter, Ramanujan defined four third order mock theta functions, ten fifth

order mock theta functions and three seventh order mock theta functions. Watson [20]

found three more mock theta functions of order three. Ramanujan gave more mock

theta functions in his lost notebook [16]. Gordon and McIntosh [9] also found mock

theta functions of order eight. Ramanujan also explained in the letter what he meant

by a mock theta function. The definition [5, p. 43] :

It is a function defined by a q-series which converges for and which

satisfies the following two conditions:

(0) For every root of unity there is a -function such that the difference

is bounded as radially.

(1) There is no single -function which works for all i.e., for every -function

there is some root of unity for which difference is unbounded as

radially.

Unfortunately no one has ever proved that mock theta functions exist.

Recently Andrews, in his path breaking paper [3], considered the role of the q-

orthogonal polynomials and showed how they reveal new mock theta functions. He

found the following mock theta functions interesting:

Two-Variable Generalization of New Mock Theta Functions 391

Bringmann, Hikami and Lovejoy [14], working on unified Witten-Reshetikhin-

Turaev invariants of certain Scifert manifolds, also found the following two new

mock theta functions:

The main aim is to prove that these functions are bounded near certain roots of

unity. We also generalize these functions and gave the functional equations satisfied

by these generalized functions and then show that they satisfy Ramanujan’s

description of mock theta functions.

The paper is organized like this. In section 2 we give a two variable

generalization of the above functions and give functional equations satisfied by these

functions. In section 3 we prove by direct application of mathematical induction that

they satisfy Ramanujan’s description of mock theta functions. Lastly section 4

contains the main theorems that ,

, and

are bounded near

certain roots of unity.

In this paper we will use the standard notations for the q-shifted factorial:

For ,


When k =1, it is customary to write instead of .

2. Two Variable Generalization (a) We first define the generalizations of mock theta functions of Andrews by

Replacing z by q in the above we see they become mock theta functions of

Andrews.

Functional Equations We now show by simple calculation that the function

satisfies the functional equation

(2.5)


Proof.

The functional equations for other functions are

and

The proofs for these functional equations are similar, so omitted.

,

, ,

are mock theta functions,

positive integer.


By using the functional equations for these generalized functions we show that

for any positive integer m these functions are satisfying Ramanujan’s description of

mock theta functions. We give the proof for only, the proof for the other

functions are similar, so omitted.

Proof. Let in the functional equation (2.5) and assume that it is true for m i.e.

Now we have to prove that the functional equation holds for also.

Putting the value of from in we get

Now we have to prove (2.11). The left side


Thus (2.11) is true. Since for

is a mock theta function,

hence by mathematical induction are mock theta functions.

(b) We now define the generalizations of the mock theta functions of Bringmann

et al by

When , the two functions above are mock theta functions of Bringmann et

al. By simple calculation the functions and

satisfy the functional

equation:


and

By using the functional equation and employing mathematical induction, we

show that for any positive integer m, and

satisfy Ramanujan’s

description of mock theta functions. The proofs being similar, so omitted.

3. The behaviour of the mock theta functions in the

neighbourhood of the unit circle

We show that the functions

and

are bounded near

certain roots of unity.

Theorem 1. Let be a rational number, expressed in their lowest form where M

is even and N is odd and for , then and

are

bounded

Theorem 2. Let N be a positive integer, a primitive root of unity, and

for , then if N is even, and are bounded for

We shall require the following Lemmas [5, Lemma 5.1 and Lemma 5.3, p. 93] in

the proof of

the Theorems

Lemma 1. Suppose that, for each , is a bounded function for Suppose further that there exist integers , and and a positive real number

such that

for all and . Then

converges and is bounded for .


Lemma 2. Let and Then

Lemma 3. If , and , then

Proof of Theorem 1. Let with M even and N odd, and put

so

Now

We apply Lemma 2 with

to get


Since

runs twice through the roots of

We thus have

.

So

Now each term of the finite sum on the right side is bounded since r lies in the

closed interval and so is bounded.

Now

where


Since and N being odd, the multiplicative order of is odd,

there exists such that

Hence

and so is also bounded.

Proof of Theorem 2. By definition

Let

then

since .

Let us assume that . We apply Lemma 2 with and


As p ranges from 1 to 2N, runs through the roots of the

polynomial

The product of these roots is multiplied by the coefficient of in (3.3) i.e.

so

Hence

Applying Lemma 3 in (3.4), we have

provided , and Obviously (3.5) is true for . Hence, by

Lemma 1, is bounded for It follows that

is bounded.

Now


The first series on the right side is , which is bounded (just proved) and the

terms of the series defining do not exceed in absolute value the corresponding

terms of the second series on the right. Hence is also bounded.

References

[1] G. E. Andrews, An introduction to Ramanujan’s “lost” notebook, Amer. Math.

Monthly, 86 (1979), 89-108.

[2] G. E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer.

Math. Soc., 293 (1986), 113-134.

[3] G. E. Andrews, q-orthogonal polynomials, Rogers-Ramanujan identities, and

mock theta functions, Proc. Steklov Inst. Math.(to appear).

[4] G. E. Andrews and F.G. Garvan, Ramanujan’s “lost” notebook VI: The mock

theta conjectures, Adv. Math.,73(1989), 242-255.

[5] G. E. Andrews, and D. Hickerson, Ramanujan’s “lost” notebook, VII: The sixth

order mock theta functions, Adv. Math., 89 no.1 (1991), 60-105.

[6] Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook,

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[8] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University

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[9] B. Gordon and R. J. McIntosh, Some eighth order mock theta functions, J.

London Math.Soc., 62 no. 2 (2000), 321-335.

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[11] D. Hickerson, On the seventh order mock theta functions, Invent. Math. 94

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[12] D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988),

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[13] R. J. McIntosh, The H and K family of mock theta functions, Canad. J. Math., 64

no. 4 (2012), 935-960.

[14] E. Mortenson, On three third order mock theta functions and Hecke-type double

sums, Ramanujan J. (to appear).

[15] S. Ramanujan, Collected Papers, Cambridge University Press, 1927, reprinted by

Chelsea, New York, 1962.

[16] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New

Delhi, 1988.

[17] Bhaskar Srivastava, Some new mock theta functions, Math. Sci. Res. J., 15 no. 8

(2011), 234- 244.

[18] Bhaskar Srivastava, A Study of New Mock Theta functions, Tamsui Oxford

Journal of information and Mathematical Sciences (to appear).

[19] Bhaskar Srivastava, A study of Bilateral new mock theta functions, American

Journal of Mathematics and Statistics,2 no. 4 (2012), 64-69.

[20] G. N. Watson, The final problem: An account of the mock theta functions, J.

London Math. Soc., 11 (1936), 55-80.

[21] G. N. Watson, The mock theta functions (2), Proc. London Math. Soc. (2) 42

(1937), 274-304.

[22] D. Zagier, Ramanujan’s mock theta functions and their applications (after

Zwegers and Ono-Bringmann), S eminaire Bourbaki. Vol. 2007/2008.

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two-variable generalization of new mock theta...

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