21-08-2014, 02:52 PM
TRANSFORMATIONS OF q - HYPERGEOMETRIC SERIES USING MOCK-THETA FUNCTIONS OF ORDER SEVEN & TEN PROJECT REPORT
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ABSTRACT
In this paper, making use of certain known identities and mock-theta functions and partial mock-theta functions of order three and five, an attempt has been made to established transformations for basic hypergeometric series.
KEY WORDS: Transformation, q-hypergeometric series, mock-theta funtion.
2000 AMS SUBJECT CLASSIFICATION : 33A30, 33D15, 11A55, secondary 11F20
Introduction, Notations and Definitions
Throughout this paper we shall adopt the following notations and definitions.
For any numbers a and q real or complex and|q| <1,
[α;q]_n= [α]_n={█((1-α)(1-αq)(1-αq^2 )…(1-αq^(n-1) );n>0@1 ; n=0)┤ (1.1)
Accordingly, we have
[α;q]_∞= ∏_(r=0)^∞▒[1-αq^r ]
Also, [a_1,a_2,a_3…a_r;q]_n=[a_1;q]_n [a_2;q]_n [a_3;q]_n… [a_r;q]_n
Following Gasper and Rahman [2], we define a basic hypergeometric series,
〖(_r^)Φ〗_s [█(a_1,a_2,a_3…a_r;q;z@b_1,b_2,b_3…〖b 〗_s )]=∑_(n=0)^∞▒([a_1,a_2,a_3…a_r;q]_n z^n)/[q,b_1,b_2,b_3…〖b 〗_s;q]_n {(-1)^n q^(n(n-1)/2) }^(1+s-r)
(1.2)
Where 0<|q|<1 and r<s+1
In this paper we have established certain transformation formulae for basic hypergeometric functions by make use of summations of truncated series and following identity,
(1.3)
where,
[Agarwal, R. P.II;Eq.10,p.79 ]
In this paper, we shall make use of the identity (1.3) in order to establish new representations of mock theta function of order seven, ten are discussed.
Mock theta functions Partial mock theta functions
of order seven. of order seven.
(i) (1.4)
(ii) (1.5)
(iii) (1.6)
Mock theta functions Partial mock theta functions
of order ten. of order ten.
(i) (1.7)
(ii) (1.8)
(iii) (1.9)
(iv)
Main Results
In this section we shall establish new representations of mock theta functions of order seven and ten.
(i) Taking in (1.3) and using (1.4), we get.
(2.1)
(ii) Taking in (1.3) and using (1.5), we get.
(2.2)
(iii) Taking in (1.3) and using (1.6), we get. (2.3)
(iv) Taking in (1.3) and using (1.7), we get.
(2.4)
(v) Taking in (1.3) and using (1.8), we get.
(2.5)
(vi) Taking in (1.3) and using (1.9), we get.
(2.6)
(vii) Taking in (1.3) and using (1.10), we get.