A Study of Bilateral New Mock Theta Functions

We define bilateral series for two sets of new mock theta functions-one given by Andrews and the other by Bringmann et al. Not only the bilateral form of the mock theta functions in the two sets are equivalent they further come out to be equivalent to the bilateral form of the eighth order mock theta functions of Gordon and McIntosh. These bilateral series are expressed as a q-hypergeometric 1 2φ series and then represented by continued fractions. We extend the definition of the bilateral mock theta functions and show they are q F -functions and relations are then defined.


Introduction
In his last letter to G.H. Hardy [11], S. Ramanujan listed seventeen mock theta functions of order three, five, five and seven. According to Ramanujan a mock theta function is a function ( ), f q 1, q < satisfying the following two conditions: (0) For every root of unity ζ , there is a θ -function ( ) q ζ θ such that the difference ( ) ( ) f q q ζ θ − is bounded as q → ζ radially.
(1) There is no single θ -function which works for all ζ i.e., for every θ -function ( ) q θ there is some root of unity ζ for which ( ) ( ) f q q θ − is unbounded as q → ζ radially.
G.N. Watson [17] found three more mock theta functions of order three. In his "Lost" Notebook Ramanujan gave six more mock theta functions which were called by G.E. Andrews and D. Hickerson [5] of order six and four mock theta functions which were called by Choi [6] of order ten. B. Gordon and R.J. McIntosh [8] generated eight mock theta functions and called them of order eight, but four of them were later found of lower order. Hikami [9] found one more mock theta function of order two.
Recently Andrews [2] in his paper generated some new mock theta functions and found four of them interesting. Bringmann, Hikami and Lovejoy developed two more mock theta functions. We in [12][13][14] have made a comprehensive study of these mock theta functions.
After studying these mock theta functions in [12][13][14] I started considering their bilateral form. The study became interesting as in their bilateral form the mock theta functions We now write the mock theta function as basic bilateral series, and following Watson, call them 'Complete'. The scheme of the paper is as follows: In section 3, we give an expansion for these functions using Slater's expansion formula [7] and show their relationship with other bilateral mock theta functions.
In section 4 and section 5, we express these 'complete' mock theta functions as 1 2 ϕ series.
In section 6, we represent these 'complete' mock theta functions as continued fractions.
In section 8, relationship between two sets of bilateral mock theta functions is shown.
In section 9 and 10, we define extended form for the 'complete' mock theta functions and show that they are q Ffunctions and certain relationship between these functions.

Basic Standard Results
We shall use the following usual basic hypergeometric notations:   We will now specialize the paramaeter in (4.1) to get expansions of bilateral mock theta functions as q-hypergeometric series.

Bilateral or 'Complete' Mock Theta Functions as Continued Fractions
We first write the bilateral 0, ( ) Dividing the expressions on both sides of (6.2) by the first summation on the right side of (6.2), we have

Relationship between Bilateral Mock Theta Functions
From these alternative definition it is interesting to note that bilateral mock theta functions developed by Andrews can be expressed as bilateral mock theta functions developed by Bringmann et al.  S q S q T q and 1 ( ) T q are eighth order mock theta functions given by Gordon and McIntosh [8].