IIJSRST11737150 | Received : 10 Oct 2017 | Accepted : 25 Oct 2017 | September-October-2017 [(3) 7: 789-793]
© 2017 IJSRST | Volume 3 | Issue 7 | Print ISSN: 2395-6011 | Online ISSN: 2395-602X
Themed Section: Science and Technology
789
Properties of Certain Bilateral Mock Theta Functions-IV
Dr. Mohammad Ahmad
Department of Mathematics National Defence, Academy Khadakwasla Pune. Maharastra, India
ABSTRACT
Bilateral mock theta functions were obtained and studied in [23]. We express them in terms of Lerch's
transcendental function f(x, ξ; q, p) We also express some bilateral mock theta functions as sum of other mock theta
functions. We generalize these functions and show that these generalizations are
functions. We give an integral
representation for these generalized functions.
Keywords: Mock theta functions, bilateral mock theta functions, Lerch transcendent, F-function.
I. INTRODUCTION
The mock theta functions were first introduced by
Ramanujan [3] in his last letter to G. H. Hardy in
January 1920. He provided a list of seventeen mock
theta functions and labelled them as of third, fifth and
seventh order without mentioning the reason for his
labelling. Watson [18] added to this set three more third
order mock theta functions.
His general definition of a mock theta function is a
function defined by -series convergent when
|| which satisfies the following two conditions.
For every root of unity, there exists a theta
function
1
such that the difference between
and
is bounded as radially.
There is no single theta function which works for all
i.e. for every theta function
there is some root
of unity for which minus the theta function
is unbounded as radially.
Andrews and Hickerson [15] announced the existence of
eleven more identities given in the ‘Lost’ note book of
Ramanujan involving seven new functions which they
labelled as mock theta functions of order six.
1
When Ramanujan refers to theta functions, he means sums,
products, and quotients of series of the form ∑
with and .
Y. S. Choi [1] has discovered four functions which he
called the mock theta function of order ten. B. Gordon
and R. J. McIntosh [30] have announced the existence of
eight mock theta functions of order eight and R. J.
McIntosh [5] has announced the existence of three mock
theta functions of order two.
Hikami [13], [14] has introduced a mock theta function
of order two, another of order four and two of order
eight. Very recently Andrews [16] while studying -
orthogonal polynomials found four new mock theta
functions and Bringmann et al [12] have also found two
more new mock theta functions but they did not mention
the order of their mock theta functions.
Watson [19] has defined four bilateral series, which he
has called the ‘Complete’ or Bilateral forms for four of
the ten mock theta functions of order five. Further he has
expressed them in terms of the transcendental function
studied by M. Lerch [7]. S. D. Prasad [2] in
1970 has defined the ‘Complete’ or ‘Bilateral’ forms of
the five generalized third order mock theta functions.
The ‘Complete’ sixth order mock theta functions were
studied by A. Gupta [31]. Bhaskar Srivastava
[26],[27],[28],[29] have studied bilateral mock theta
functions of order five, eight, two and new mock theta
functions by Andrews [6] and Bringmann et al [12].
Truesdell [25] calls the functions which satisfy the
equation
as - functions. He
has tried to unify the study of these -functions. The