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Two-variable generalization of new mock theta functions.

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 [y.sub.0]u 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 f(q) defined by a q-series which converges [absolute value of q] < 1 for and which satisfies the following two conditions:

(0) For every root of unity [zeta] there is a [theta]-function [theta] [sub.[zeta]](q) such that the difference f(q) -[theta] [sub.[zeta]](q)is bounded as q [right arrow] radially.

(1) There is no single [theta]-function which works for all [zeta] i.e., for every [theta]-function [theta](q) there is some root of unity [zeta] for which difference f(q) - [theta](q) is unbounded as q [right arrow] [zeta] 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 qorthogonal polynomials and showed how they reveal new mock theta functions. He found the following mock theta functions interesting:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

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:

[[bar.[phi]].sub.[bar.0]] = [[infinity].summation over (n = 0)] [q.sup.n] [(-q; q).sub.2n+1] (i-5)

and

[[bar.[phi]].sub.[bar.1]] = [[infinity].summation over (n = 0)] [q.sup.n] [(-q; q).sub.2n]. (1.6)

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 [[bar.[psi]].sub.[bar.0]](q), [[bar.[psi]].sub.[bar.1]](q), [[bar.[phi]].sub.[bar.0]](q), and [[bar.[phi]].sub.[bar.1]](q) are bounded near certain roots of unity.

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

For [absolute value of [q.sup.k]] < 1

[(a; [q.sup.k]).sub.0] = 1,

[(a; [q.sup.k]).sub.n] [[PI].sup.n-1.sub.m=0](1 - [aq.sup.km]), n [greater than or equal to] 1

[(a; [q.sup.k]).sub.[infinity]] [[PI].sup.[infinity].sub.m=0](1 - [aq.sup.km])

When k = 1, it is customary to write [(a).sub.n] instead of [(a; q).sub.n].

2. Two Variable Generalization

(a) We first define the generalizations of mock theta functions of Andrews by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

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 [[bar.[phi]].sub.[bar.0]] (z,q) satisfies the functional equation

[[bar.[phi]].sub.[bar.0]] (z,q) = [z.sup.4]/[q.sup.2](1 + [z.sup.2]) (1 + [z.sup.2]/q) [[bar.[phi]].sub.[bar.0]](zq,q) + 1. (2.5)

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The functional equations for other functions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.7)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.8)

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

[[bar.[psi]].sub.[bar.0]]([q.sup.m],q), [[bar.[psi]].sub.[bar.1]]([q.sup.m],q), [[bar.[psi]].sub.[bar.2]]([q.sup.m],q), [[bar.[psi]].sub.[bar.3]]([q.sup.m],q) are mock theta functions, m 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 [[bar.[psi]].sub.[bar.0]]([q.sup.m],q) only, the proof for the other functions are similar, so omitted.

Proof. Let z = [q.sup.m] in the functional equation (2.5) and assume that it is true for m i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Now we have to prove that the functional equation holds for z = [q.sup.m+1] also.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

Putting the value of [[bar.[psi]].sub.[bar.0]] ([q.sup.m+1],q) from (2.9) in (2.10) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Two Variable Generation of New Mock Theta Functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus (2.11) is true. Since for m = 1, [[bar.[psi]].sub.[bar.0]](q, q) = [[bar.[psi]].sub.[bar.0]] (q) 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

[[bar.[phi]].sub.[bar.0]] (z,q) = [[infinity].summation over n = 0][q.sup.n][(- [z.sup.2]/q;q).sub.2n+1] (2.12)

[[bar.[phi]].sub.[bar.1]] (z,q) = [[infinity].summation over n = 0][q.sup.n][(- [z.sup.2]/q;q).sub.2n+1] (2.13)

When z = q the two functions above are mock theta functions of Bringmann et al. By simple calculation the functions [[bar.[phi]].sub.[bar.0]] (z,q) and [[bar.[phi]].sub.[bar.1]] (z,q) satisfy the functional equation:

[[bar.[phi]].sub.[bar.0]](zq,q) = [1/q(1 + [z.sup.2])(1 + [z.sup.2]/q)][ [[bar.[phi]].sub.[bar.0]](z,q) - (1 + [z.sup.2]/q)]

and

[[bar.[phi]].sub.[bar.0]](zq,q) = [1/q(1 + [z.sup.2])(1 + [z.sup.2]/q)][ [[bar.[phi]].sub.[bar.0]](z,q) - 1].

By using the functional equation and employing mathematical induction, we show that for any positive integer m, [[bar.[phi]].sub.[bar.0]]([q.sup.m],q) and [[bar.[phi]].sub.[bar.0]]([q.sup.m] ,q) 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 [[bar.[psi]].sub.[bar.0]](q), [[bar.[psi]].sub.[bar.1]](q) and [[bar.[phi]].sub.[bar.0]](q) [[bar.[phi]].sub.[bar.1]](q)are bounded near certain roots of unity.

Theorem 1. Let M/N be a rational number, expressed in their lowest form where M is even and N is odd and q = [re.sup.M[pi]/N] for 0 [less than or equal to] r [less than or equal to] 1, then [[bar.[psi]].sub.[bar.0]](q) and are [[bar.[psi]].sub.[bar.1]](q) bounded

Theorem 2. Let N be a positive integer, [zeta] a primitive Nth root of unity, and q = r[zeta] for 0 [less than or equal to] r [less than or equal to] 1, then if N is even, [[bar.[phi]].sub.[bar.0]](q) and [[bar.[phi]].sub.[bar.1]](q) are bounded for 0 [less than or equal to] r [less than or equal to] 1.

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 n [greater than or equal to] 0, [u.sub.n] (r) is a bounded function for a [less than or equal to] r [less than or equal to] b. Suppose further that there exist integers N [greater than or equal to] 1, and K [greater than or equal to] 0 and a positive real number [alpha] < 1 such that

[absolute value of [u.sub.n+N](r)] < [alpha][absolute value of [u.sub.n](r)]

for all n [greater than or equal to] K and a [less than or equal to] r [less than or equal to] b. Then

[[infinity].summation over n =0] [absolute value of [u.sub.n](r)]

converges and is bounded for a [less than or equal to] r [less than or equal to] b.

Lemma 2. Let 0 < R' [less than or equal to] R [less than or equal to] 1 and [absolute value of z] [less than or equal to] 1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 3. If a > 0, b > 0 and 0 [less than or equal to] r [less than or equal to] 1, then

[r.sup.a](1 - [r.sup.b]) < [b/a + b]

Proof of Theorem 1. Let q = [r.sup.eM[pi]/N] with M even and N odd, and put n = nN + m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so

[[u.sub.n,m]/[u.sub.n+1,m]] = [q.sup.-2N(N+2nN+2m)][(-[q.sup.-2nN+2nN+2m+1];q).sub.2N]

Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We apply Lemma 2 with R = [r.sub.2mN+2m+1], R' = [r.sub.2mN+2m+p] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] runs twice through the roots of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We thus have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now each term of the finite sum on the right side is bounded since r lies in the closed interval 0 [less than or equal to] r [less than or equal to] 1 and so [[bar.[psi]].sub.[bar.0]](q) is bounded.

Now

[[bar.[psi]].sub.[bar.0]](q) = [[infinity].summation over n=0][[q.sup.2n]/1+[q.sup.2n+1]][u.sup.n](r),

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since q = [re.sup.M[pi]i/N] and N being odd, the multiplicative order of [e.sup.M[pi]i/N] is odd, there exists c > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence

[absolute value of [q.sup.2n]/1 + [q.sup.2n+1]] [less than or equal to] 1/c

and so [[bar.[psi]].sub.[bar.1]](q) is also bounded.

Proof of Theorem 2. By definition

[[bar.[phi]].sub.[bar.1]](q) = [[infinity].summation over n=0 [(-q; q).sub.2n][q.sup.n] (3.1)

Let

[V.sub.n] (r) = [q.sup.n] [(-q; q).sub.2n]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

since q = r [zeta].

Let us assume that 0 < r [less than or equal to]. We apply Lemma 2 with R = [r.sup.2n+p], R' = [r.sup.2n+N/2] and z [[zeta].sup.2n+p]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As p ranges from 1 to 2N, 1 + [r.sup.2n+N/2] [[zeta].sup.2n+p] runs through the roots of the polynomial

[(x - 1).sup.2N] [r.sup.2N(2n+N/2)] (3.3)

The product of these roots is [(-1).sup.N] multiplied by the coefficient of [x.sup.0] in (3.3) i.e.

1 - [r.sup.2N(2n+N/2)]

so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Applying Lemma 3 in (3.4), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

provided n [greater than or equal to], and 0 < r [less than or equal to] 1. Obviously (3.5) is true for r = 0. Hence, by Lemma 1, [[SIGMA].sup.[infinity].sub.n=0][absolute value of [V.sub.n](r)] is bounded for 0[less than or equal to] r [less than or equal to] 1. It follows that [[bar.[phi]].sub.1] = [[SIGMA].sup.[infinity].sub.n=0][V.sub.n](r) is bounded.

Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first series on the right side is [[bar.[phi]].sub.1](q), which is bounded (just proved) and the terms of the series defining [[bar.[phi]].sub.0](q) do not exceed in absolute value the corresponding terms of the second series on the right. Hence [[bar.[phi]].sub.0](q) is also bounded.

Received June 15, 2013, Accepted July 26, 2013

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, Invent. Math. 136 no.3 (1999), 497-569.

[7] Y.-S. Choi, Tenth order mock theta functions in Ramanujan's 'Lost' Notebook II., Adv. Math., 156 no.2 (2000), 180-285.

[8] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.

[9] B. Gordon and R. J. McIntosh, Some eighth order mock theta functions, J. LondonMath.Soc, 62 no. 2 (2000), 321-335.

[10] B. Gordon and R.J. McIntosh, A survey of classical mock theta functions, Partions, q-series, and Modular forms, Develop. Math., 23 (2012), 95-144.

[II] D. Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), 661- 677.

[12] D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639-660.

[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 andStatistics,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), Seminaire Bourbaki. Vol. 2007/2008. Asterisque no. 326(2009),Exp. no. 986, vii-viii, 143-164 (2010).

[23] S. P. Zwegers, Mock theta functions, Ph. D. Thesis, Universiteit Utrecht, 2002.

Bhaskar Srivastava ([dagger])

Department of Mathematics and Astronomy Lucknow University Lucknow, India

* 2000Mathematics Subject Classification. Primary 33D15.

([dagger]) Email: bhaskarsrivastav@yahoo.com
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