Vol 2, No 12 (2011)

(1)

!!!" #

"

ON CERTAIN SUMMATION AND TRANSFORMATION FORMULAS

FOR BASIC HYPERGEOMETRIC SERIES

Soni Singh*

Department of mathematics, T. D. P .G. College Jaunpur (India) E-mail: [email protected]

(Received on: 03-09-11; Accepted on: 17-12-11)

________________________________________________________________________________________________ ABSTRACT

In this paper making use of Bailey’s transform and certain known summations; an attempt has been made to establish interesting transformation formulas for basic Hypergeometric series and. Summation formula.

Keywords and phrases: Transformation, summation formulae, Basic Hypergeometric series.

________________________________________________________________________________________________

INTRODUCTION, NOTATIONS AND DEFINITIONS:

W.N. Bailey in 1944 established the following result: (1 ;( 2.4.1), (2.4.2) & (2.4.3) p.58, 59) If

0

n r n r n r

r

u

v

β

α

₋ ₊ =

∞

=

(1.1)

And

2

r r r n r n r n r r n

r n r n

u

v

u v

γ

δ

∞ ∞

− + + +

= =

=

(1.2)

then under suitable convergence conditions

0 0

n n n n

n n

α γ

β δ

∞ ∞

= =

=

(1.3)

where ,

α δ

_r_, _r_,

u v

_r

,

_r,are the functions of r only, such that the series for

γ

_rexists. This transformation lead to various results, which play important roles in number theory and transformation theory of hypergeometric series, we show that this transform can be utilized to establish certain transformations of ordinary hyper geometric series.

The basic Hypergeometric series is defined as,

( 1) / 2 1 , ,..., ; (( 1) ) , ,..., ; ;

1 2 1 2

, , ,...., ₀ _, _, _,..., _;

1 2 _{1 2}

n n

n n s r

a a a q z q

a a a q z

r

r n

r s b b b _n _{q b b} _{b q}

s _s

n

− + −

− ∞

Φ =

=

(1.4)

where

1

,

2

,...,

r

;

_n 1

;

_n 2

;

_n

...

r

;

_n

,

a a

a q

=

a q

(1.5a)

(

)(

)

(

1

)

;

1

1 ... 1

n

,

n

a q

= −

a

−

aq

−

aq

−

(1.5b) (1.5c)

and

(

)

0

1 .

;

r

aq

a q

∞ ∞

=

−

=

_∏

(1.5d)

(2)

2 3 " 3 &

The series (1.4) converges for IqI<1, IzI<1 when r =s+1 and for all value of z when r s. when r>s+1, the series diverges for all z 0 .

We shall use the following known sums due to Verma and Jain (2) in the next section.

2 2

2 2 2

2 ,

,

_{( 1) ( ; ) (}

_{; ) (}

_;

_{) (}

_;

₎

; ;

4 3

1

1 _{( ; ) ( ; ) (}

2 2 2

_;

2 _{) (}

2 _;

2 ₎

,

n

q

n

q

_xy

xq yq

_{q q}

_{xyq q}

_{x q}

_q

_{y q}

_q

n

m

q q

n

q

_q

_{x q}

_{y q}

_{x y q}

_q

xyq

_x

_y

n

m

−

− −

₋

Φ

₋

=

−

. (1.6)

where m is greatest integer n/2. (2 ; (2.10) p-1025)

(

1)/2

,

₂

,

_{( 1) ( ; ) (}

2 2

_;

2 ₎

2

2 ; ;

2 2

1

1 ₍

2 _;

2 _{) (}

2 _;

2 ₎

,

n

q

n

_{n n}

q

n

_{q q}

_{x q}

_q

_x

n

m

n

m

x

_q

n

q

_x

_q

m

x

−

₋

−

Φ

−

=

−

(1.7)

where m is greatest integer n/2.(2 ; (2.19) p-1026)

,

_{( 1) ( ; ) (}

2 2

_;

2 ₎

(

1)/2

2 ; ;

2 2

1 _{( ; ) (}

_{; ) (}

2 _;

2 ₎

,

n

q

n

q

n

_{q q}

_{x q}

_q

n n

_x

n

m

x

_q

n

q

_{x q}

_{xq q}

_q

n

m

x

−

₋

−

Φ

−

=

−

₋

−

(1.8)

2 2

2 2 2

2 2 2 1

( ; ) (

;

) (

;

)

,

, ,

_{; ;}

4 3

_,

2 ₍

2 _{; ) (}

2 2 2

_;

2 _{) (}

2 _;

2 ₎

n

x

q q

x q

q

y q

q

x y q

x xq

_{q q}

_n

_m

xyq xyq x q

_{x q q}

_{x y q}

_q

n

m

−

+

₋

Φ

=

−

(1.9)

2 2 4 2 2

_,

2 2

_,

2 ₂

₂

(

; ) (

; )

;

4 3

2 _,

2 2 4 2

_,

₍

2 _{; ) (}

2 _{; )}

n

x

q q

bq q

b x q

x x q q

_q

n

x bq bx q x q

_{bx q q}

_{x q q}

n

+

−

Φ

=

−

(1.10)

2

2 2 3

2

2 ( ; ) (

; ) (

;

) (

;

) (

; )

2 2

,

, ,

_{; ;}

2 4 3

2 2

_,

₍

_{; ) (}

2 2

_{; ) (}

2 _;

2 _{) (}

2 _{; )}

₍

2 3

_;

2 ₎

2 n

x

q q

bxq

q

bq

q

bx q

q

xq

q

n

_n

_m

q

bx q

x

xq

_{q q}

m

x q

xq b xq b

_{xq q}

_{bx q}

_q

_bxq

_q

_{x q}

_q

n

m

_m

m

−

+

₋

Φ

=

−

(1.11)

2

1 1/2

1/2

( ; ) (

; ) (

;

)

,

, , ( )

,( )

; ;

5 4

₍

₎

1/2

_{, (}

₎

1/2

_,

_{, /}

₂

(

;

)

(

;

) (

; ) (

; )

a

n m

x

n

x

q q

aq

q

xq q

q

aq

x

_q

_n

_m

x

q q

a

m

aq

xq x q

_q

_{q q}

_{aq q}

_{xq q}

m

_x

m

n

−

+

₋

Φ

=

−

(1.12)

3

3 _,

3 3 3

_{, ,}

_,

2 _,

₃

(

; ) (

;

)

;

5 4

₍

₎

3/2

_{, (}

₎

3/2 3/2 3

_,

3/2 3

₍

3 3

_;

3 _{) ( ; )}

n

a

aq q

q

a q

a aq aq

_n

q

aq

a

q

a

q

_{a q}

_q

_{q q}

n

−

+

Φ

=

(3)

2 3 " 3 &

2 4

3 3 6

3 ( ; ) (

; ) (

; )

(

;

)

3 4

2 ,

, ,

,

_{; ;}

3 5 4

₍

₎

3/2

_{, (}

₎

3/2 3/2 2

_,

3/2 2

₍

3 _;

3 _{) (}

2 4

_{; )}

₍

3 4

_{; ) (}

_{; )}

3 n

x

q q

x q

q

xq

q

x q

q

n

m

q

x q

x wxq w xq

_{q q}

m

xq

x

q

x

q

_q

_{x q}

_q

_{x q}

_q

_{xq q}

m

_m

n

−

+

Φ

=

−

(1.14)

3

1 1/3 1/3

1/3 2

(

)

( ; ) (

;

)

,

; ;

5 4

₍

₎

1/2

_{, (}

₎

1/2

_,

1/2 1/2

_,

₍

3 _;

3 _{) (}

_{; )}

n m

n

a

q q

aq

q

aq

a

w a

w

_n

_m

q q

aq

a

q

_q

_{aq q}

m

n

−

+

Φ

=

−

(1.15)

3 3 6 3 1 1/3 1/3 1/3 2 ( ) ( ; ) ( ; ) ( ; ) ( ; ) , , , , ,

; ; 5

6 ₍ ₎1/2_{, (} ₎1/2_, 1/2 1/2 2_, _, ₍ 2 _{; ) (} _{; ) (} 3_; 3_{) (} _; 3₎

n m

n n a q q a q aq q q a q

q a q aq a a w a w _{q q} n n m m

aq aq a a q a _q _{a q} _{aq q} _q _q _{a q}

n n m m

−

− +

Φ =

− −

(1.16)

2- MAIN RESULTS:

In this section we shall establish our main results .

(1) Choosing r=

[

; ] [

; ]

[

; ] [ ; ]

r r

xq q

yq q

xyq q

q q

−

, ur=

[ ; ] [ ; ]

(

)

[

; ] [ ; ]

r r r

r r

x q

y q

q

xyq q

q q

−

, vr=1, and r=1 in (1.1) (1.2) and making use of

(1.6) we get

n=

2 2

2 2 2

2 [

; ] [

;

] [

;

]

2 2 2

2

2 [

; ] [

;

] [

;

]

xyq q

_n

x q

q

_m

y q

q

_m

xyq q

_n

x y q

q

_m

q

_m

−

, (2.1)

where m is greatest integer n/2 and,

n=

[

; ] [

; ]

[

; ] [

; ]

xq q

yq q

xyq q

q q

∞ ∞

−

. (2.2)

Putting these values in (1.3) we get

2 2 2 2 2

,

_;

,

_{; ;1}

,

_;

2 _;1

,

_{2 1}

_{4 3}

_,

_{2 2 2 2}

_,

xq

yq

_q

x y

_q

xyq xyq x q

y q

_q

xyq q

xyq

_xyq

_{xyq x y q}

−

Φ

= Φ

∏ −

−

₋

+

2 3 2 2 2 2

(1

)

,

_;

2 _;1

4 3

2 3 2 2 2

(1

)

,

xyq

xyq xyq x q y q

q

xyq

xyq x y q

−

Φ

+

−

(2.3)

where either x or y is of the form of q-n.

(2) Setting r=

(

1)/2

[ ; ]

r r

q

q q r

−

, ur=

[ ; ] [

; ]

( 1)

2 [

; ] [ ; ]

x q

_r

x q

_r

x q q

_r

q q

_r

−

, vr=1, and r= qr in (1.1) ,(1.2) and using (1.7) we get

n=

(

1)/2

2 2

2

2 [

;

]

2

2 [

; ] [

;

]

n n

n

m

x q

q

_m

q

x

x q q

_n

q

_m

−

, (2.4)

n=

[

; ] [

; ]

2 [

; ] [

; ]

n

xq q

q

x q q

q q

−

∞

−

∞

(4)

2 3 " 3 & 4

2 _[

_{; ] [}

_{; ]}

2 2 3

2 2 5

;

0 1

2 ₍₁

2 ₎

0 1

2 3

_[

2 _{; ]}

xq q

x q

q

x q

q

x q

_{x q}

x q

_{x q q}

−

∞

Φ

+

Φ

=

−

_∞

. (2.6)

(3) Again setting r=

(

1)/2

[ ; ]

r r

q

q q r

−

, ur=

[ ; ] [

; ]

( 1)

2 [

; ] [ ; ]

x q

_r

xq q

_r

x q q

_r

q q

_r

−

, vr=1, and r=1 in (1.1) (1.2) and making use

of (1.8) we have

n=

(

1)/2

2 2

2 [

;

]

2

2 [

; ] [

;

]

n n

n

x q

q

_m

q

x

x q q

_n

q

_m

−

, (2.7)

n=

[

; ] [

; ]

2 [

; ] [ 1; ]

xq q

x q

x q q

q

−

∞

−

∞

. (2.8)

[

; ] [

; ]

2

2 2 3

;

0 1

2 ₍₁

2 ₎

0 1

2 3

_[

2 _{; ]}

xq q

x q

x

q

x q

q

x q

_{x q}

x q

_{x q q}

−

∞

Φ

+

Φ

=

−

_∞

. (2.9)

(4) Choosing r=

/2

[ ; ] [

; ]

( 1)

2 [

; ] [

; ] [ ; ]

r

x q

_r

xq q

_r

q

_r

xyq q

_r

xyq q

_r

x q q

_r

q q

_r

−

, ur=

2/2

[ ; ]

r

q

q q r

, vr=[x 2

y 2 q;q]r ,and r=

2/2

r

z

r

q

in (1.1) ,(1.2) and using (1.9) we get

2 2 2

2 2

2 2 2

2 /2

[

; ] [

;

] [

;

]

2 2 2

2

2 [

; ] [

;

] [

;

]

n

n n

x y q q

_n

x q

q

_m

y q

q

_m

x q

x q q

_n

x y q q

_m

q

_m

β

= , (2.10)

where m is greatest integer n/2 and

n=

2 2

/2

2 2

_[

_{; ]}

_{( 1)}

[

; ]

₂

1 2 2

[ ; ]

_[

_{; ] [}

_{; ]}

n n

x y q q

q

x y zq q

_n

z q

_z

_{q q}

_{x y zq q}

n

−

∞

−

∞

. (2.11)

2 2

_1/2

_{2 2}

[

; ]

_,

_{2 2}

; ;

4 3

2 2 2

2 1

2 [ ; ]

,

, /

x y zq q

_{x xq xya}

_xya

_{x y q y q}

q q

q x z

z q

x q x y zq q z

x q

−

∞ Φ

= Φ

∞

2 2

_{2 2 3 2 2}

(1

)

,

_;

2 _;

2 2

2 1

2 3

2 (1

)

x y q xz

x y q y q

q

x z

x q

−

+

Φ

−

(5)

2 3 " 3 & 5

(5) Setting r=

2

2 [

;

] [

;

]

( 1)

2

2 2 2

2 4 2

2

2 [

;

] [

;

] [

;

] [

;

]

r

x

q

_r

x q q

_r

q

_r

bx q q

_r

bx q

q

_r

x q

q

_r

q

_r

−

, ur=

2

2 [

;

]

r

q

_r

, vr=[b2x4 q2 ;q2]r

and r=

2 r

z

r

q

2

2 [

; ] [

; ]

2 [

; ] [ ; ]

n

n n

bx q q

_n

bq q

_n

x

q

x q q

_n

q q

_n

β

=

−

, (2.13)

n=

2 4 2

2 2 4

2

2 _[

_;

_]

_{( 1)}

[

;

]

₂

2 1 2

2 2 4

2

2 [ ;

]

[

;

] [

;

]

n n

b x q

q

b x zq

q

_n

z q

z

q

_n

b x zq

q

_n

−

∞

−

∞

. (2.14)

2 4

2

2 _{2 2}

₂

_{2 2}

₂

[

;

]

_,

₂

_,

₂

;

; ;

4 3

4 2 2 4

2 2

2 1

2

2 _,

_,

_/

[ ;

]

b x zq

q

_{x x q bx q bx q}

_{bx q bq}

q

q x z

x q b x zq q

z

x q

z q

−

∞ Φ

= Φ

−

∞

.

(2.15)

(6) Again taking r=

/2

[ ; ] [

; ]

( 1)

2 2

[

; ] [

; ] [ ; ]

r

x q

_r

xq q

_r

q

_r

xq b q

_r

xq b q

_r

x q

q

_r

q q

_r

−

, ur=

2/2

[ ; ]

r

q

q q r

, vr=[bx 2

q2 ;q]r ,

and r=

2/2

r

z

r

q

2

2 2 3

2

2 /2

2 _[

_;

_{] [}

_;

_{] [}

_{; ]}

[

; ]

₂

2 2 3

2

2 [

; ]

_[

_{; ]}

_[

_;

_{] [}

_;

_]

2

n

n n

bq

q

bx q

q

xq

q

x q

bxq

q

_n

m

_m

xq q

_n

_bxq

_q

_{x q}

_q

_m

_q

_m

m

β

= , (2.16)

n=

2

2 2 2

/2

[

; ]

[

; ] ( 1)

1

2

2 [ ; ]

_[

_{; ] [}

_{; ]}

n n

bx zq

q

bx q

q

_n

q

z q

_z

_{q q}

_{bx zq}

_q

n

−

∞

−

∞

. (2.17)

2

2 _1/2

_1/2

[

; ]

,

(

)

,

(

)

; ;

4 3

2 2

2

2 [ ; ]

,

, /

bx zq

q

x xq xq aq

xq aq

q q

z q

x q bx zq q z

−

∞ Φ

∞

=

2 2 3

_,

2 _,

3 _{2 2}

₍₁

₎

4 _,

2 3

_,

2 ₂

_{2 2}

; ;

;

3 2

_,

2 3

₍₁

₎

3 2

2 3

_,

2 bxq

xz

bx q bq xq

bxq bx q bq

q x z

q

x z

xq x q

xq

x q bxq

−

Φ

+

Φ

−

. (2.18)

(7) Choosing r=

1/2

/2

[ ; ] [( / )

; ] [ ( / )

; ]

( 1)

[

; ] [

; ] [ / ; ] [ ; ]

r

x q

_r

x q

q

_r

x q

q q

_r

aq q

_r

aq q

_r

xq q

_r

x q q

_r

q q

_r

−

, ur=

2/2

[ ; ]

r

q

q q r

, vr=[aq;q]r , and

r=

2/2

r

z

r

q

(6)

2 3 " 3 & 6 n=

2

2 /2

[

/ ; ] [

;

] [

;

]

2

2 [

; ]

_[

_{/ ;}

_{] [}

_;

_]

n m n

aq x q

_n

aq

q

_m

xq q

_m

x

q

m

xq q

_n

_{aq x q}

_q

m

−

, (2.19)

n=

/2

[

; ]

( 1)

[

; ]

₂

1 [ ; ]

_[

_{; ] [}

_{; ]}

n n

aq q

q

azq q

_n

z q

_z

_{q q}

_{azq q}

n

−

∞

−

∞

. (2.20)

1/2

[

; ]

_,

_{,( / )}

_{, ( / )}

; ;

5 4

/ ,

,

, /

[ ; ]

azq q

_{x q a q a x q}

_{x q}

q q

x q xq azq q z

z q

−

∞ Φ

∞

=

2 _,

₍

₎

2 _,

3 _,

_,

2 _,

2

2 ;

;

2 1

2 ₍₁

₎

4 3

3 _,

_{/ ,}

2 _,

a

_{q aq}

a

_q

a

_{q xq aq}

x

x aq z

x

_q

_z

x

_q

_z

q

xq

q

xq

xq aq x xq

−

Φ

+

Φ

−

. (2.21)

(8) Setting r=

3

2

3 3 /2

[ ;

] [

;

] [

;

]

( 1)

3/2

3 3/2

3 3/2 3

3

3 [(

)

;

] [ (

)

;

] [

;

] [

;

] [

;

]

r

a q

_r

aq q

_r

aq

q

_r

q

_r

aq

q

_r

aq

q

_r

a

q

_r

a

q

_r

q

_r

−

ur=

2

3 /2

3 3

[

;

]

r

q

_{q r}

, vr=[a3q3 ;q3]r , and r=

2

3 /2

r

z

r

q

in (1.1) ,(1.2) and making use of (1.13) we get

n=

2

3 /2

[

; ]

[ ; ]

n

aq q

_n

a q

q q n

, (2.22)

n=

3 3

3 3 /2

3

3 _[

_;

_]

_{( 1)}

[

;

]

₂

3 1 3

3

3 [ ;

]

[

;

] [

;

]

n

a q

q

a zq

q

_n

z q

z

q

_n

a zq

q

_n

−

∞

−

∞

. (2.23)

3

2

2 [ ;

] [

; ]

,

_;

3 _;

3 3 2

3 3 3

_,

_/

_[

3

3 _;

3 _{] [}

_{; ]}

z q

a zq q

a aq aq

q

a zq q

z

_{a zq}

_q

_{zq q}

∞

Φ

=

∞

(2.24)

(9) Choosing r =

2 /2

[ ; ] [

; ] [

; ]

( 1)

3/2

3/2 2

[(

)

; ] [ (

)

; ] [

; ] [ ; ]

r

x q

_r

wxq q

_r

w xq q

_r

q

_r

xq

q

_r

xq

q

_r

x

q

_r

x

q

_r

q q

_r

−

,vr=[x 3

q 4; q]r

ur=

2/2

[ ; ]

r

q

q q r

, and r=

2/2

r

z

r

q

in (1.1) ,(1.2) and making use of (1.14) we get

n=

2 2 4

3 6

3

3 /2

[

; ] [

;

] [

; ]

3 2 4

3 3

[

; ] [

; ]

[

;

]

3 n n

x q

q

_n

x q

q

_m

xq

q

x q

m

xq q

_n

x q

q

_m

m

(7)

2 3 " 3 &

n=

3 4

/2

3

4 _[

_{; ]}

_{( 1)}

[

; ]

₂

1

3

4 [ ; ]

_[

_{; ] [}

_{; ]}

n n

x q

q

x zq

q

_n

z q

_z

_{q q}

_{x zq}

_q

n

−

∞

−

∞

. (2.26)

3

4 ₂

_{3/2 5/2}

[

; ]

_,

; ;

5 4

3

4 3/2

3/2

[ ; ]

/ ,

,( / )

, ( / )

x zq

q

_{x wxq xw q x}

_q

_x

_q

q q

z q

q z x zq

x q

−

∞ Φ

−

∞

=

2 4

3 6

_,

4 _,

5 _,

_{3 3 3}

₍₁

₎

2 7 3 6

_,

6 _,

_{3 3 3}

;

3 2

_,

2 ₍₁

₎

3 2

2 2 4

_,

x q

xz

x q xq xq

x q x q xq

q

x z

q

x z

xq

xq xq

xq x q

−

Φ

+

Φ

−

2 4

2 5

2 _{2 7 2 8 3 6}

(1

)(1

)(

)

,

_;

3 3 3

_;

3 2

2 4 2 5

2 _,

(1

)(1

)

x q

xz

x q x q x q

q

x z

x q x q

xq

−

+

Φ

−

(2.27)

(10) Choosing r=

1/3

2 1/3

/2

[

; ] [

; ]

( 1)

1/2

[(

)

; ] [ (

)

; ] [

; ] [ ; ]

r

a

q

_r

wa

q

_r

w a

q

_r

q

_r

aq

q

_r

aq

q

_r

a

q q

_r

a

q

_r

q q

_r

−

, vr=[aq; q]r ,

ur=

2/2

[ ; ]

r

q

q q r

and r=

2/2

r

z

r

q

n=

2

3

3 /2

[

;

] (

)

3

3 [

;

]

n m n

aq

q

_m

a

q

_{q m}

−

, (2.28)

n=

/2

[

; ]

( 1)

[

; ]

₂

1 [ ; ]

_[

_{; ] [}

_{; ]}

n n

aq q

q

azq q

_n

z q

_z

_{q q}

_{azq q}

n

−

∞

−

∞

. (2.29)

3

3 1/3

_,

1/3

_,

2 1/3

_,

₂

[ ; ] [

;

]

; ;

(1

)

4 3

1/2

_,

_{, /}

_[

3 _;

3 _{] [}

_{; ]}

z q

a zq

q

a

wa

w a

q a

q q

z a

az

a

zaq q z

_az

_q

_{azq q}

−

_∞

Φ

=

+

−

_∞

_(2.30)

(11) Choosing r=

1/3

2 1/3

/2

[

; ] [

; ]

( 1)

1/2

2 [(

)

; ] [ (

)

; ] [

; ] [ ; ]

r

a

q

_r

wa

q

_r

w a

q

_r

q a q

_r

q

_r

aq

q

_r

aq

q

_r

a

q

_r

a

q

_r

q

a q

_r

q q

_r

−

, vr=[aq; q]r

ur=

2/2

[ ; ]

r

q

q q r

, and r=

2/2

r

z

r

q

n=

2

3

6

3 /2

[

;

] [

; ] [

;

] (

)

2

3

3 1/2

3 [

; ] [

;

] [

;

]

n m n

aq

q

_m

a q

_n

q

a q

_m

a

q

a q

_n

q

_m

a

q

_m

−

, (2.31)

n=

/2

[

; ]

( 1)

[

; ]

₂

1 [ ; ]

_[

_{; ] [}

_{; ]}

n n

aq q

q

azq q

_n

z q

_z

_{q q}

_{azq q}

n

−

∞

−

∞

(8)

2 3 " 3 &

1/3

1/3 2 1/3

[

; ]

,

_{; ;}

5

6

2 [ ; ]

/ ,

,

azq q

a

wa

w a

q a q a q a

q q

z q

q z azq q

a

−

∞ Φ

−

∞

=

,

3 6

,

;

3 ;

3 (1

)

,

2 ,

3 6

,

;

3 ;

3 3 2

3 _,

4 ₍₁

2 ₎

4 3

4 _,

5 _,

a

a z

q a aq q

a

q a q

a aq q

a

q

az

q

az

q

a q

a

_{q a}

q

a q

a

−

Φ

+

Φ

−

2 ₂

₃

(1

)(1

)

,

_;

3 _;

3

5 3 2

2

3 _,

(1

)(1

)

a

q a az

q

a q

a aq

q

az

q

a

q

a

q

a

−

+

Φ

−

(2.33)

ACKNOWLEDGEMENT:

We are thankful to Dr. S.N. Singh, Department of Mathematics, T.D.P.G. College, Jaunpur for his valuable guidance in the preparation of this paper. This work has been done under the research project of DST No. SR/S4/MS: 524/08 sanctioned to Dr. Singh.

REFERENCES:

[1] Slater, L. J.: Generalized Hyper geometric Functions, Cambridge University Press (1966).

[2] Verma, And Jain V.K.: some Summation formulas of basic Hypergeometric series, India J. Pure and Apple math, 1980; 11(8):1021-1038

[3] Singh U.B.: A note on transformation of Bailey. Quart. J. Math. Oxford Ser. 1994; 45(2):111-116.

[4] Denis R. Y., Singh S.N., and Singh S.P: On Certain special transformation involving Basic Hypergeometric functions. J. of Indian Math. Soc.2010; 77:47-55.