On Transformation Formulae for Basic Hypergeometric Functions by using WP Baley’s Pairs

(1)

On Transformation Formulae for Basic

Hypergeometric Functions by using WP

Baley’s Pairs

Akash Kumar Srivastava

#1

, Neera A. Herbert*

2

#1

Sam Higginbottom University of Agriculture, Technology and Sciences, Allahabad, Uttar Pradesh, India

*2

Sam Higginbottom University of Agriculture, Technology and Sciences, Allahabad, Uttar Pradesh, India

Abstract:-

In this paper making use of well known WP Bailey’s Pairs, an attempt has been made to established

certain interesting transformation formulae for Basic Hypergeometric functions.

Key words:-

Bailey’s transform, Bailey Pairs, WP Bailey Pairs, transformations formulae.

1. Introduction

Transformation theory play very important role in the theory of q-hypergeometric series. Rogers-Ramanujan

type identities are established through transformation formulae and identities have great importance in the

theory of partitions.

By using WP Bailey’s pairs S.N. Singh [1] established transformation formulae for q-series, as

(1.1)

4

ɸ

3 𝑘,𝑘𝑦 /𝑎,𝑘𝑧 /𝑎,𝑎𝑞 /𝑦𝑧 ;𝑞;𝑎 2_𝑞/𝑘2

𝑎𝑞 /𝑦,𝑎𝑞 /𝑧,𝑘𝑦𝑧 /𝑎

=

(𝑎𝑞 /𝑘,𝑎2𝑞/𝑘;𝑞)∞ (𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞)

∞

×

10

ɸ

9

𝑎,𝑞 𝑎,−𝑞 𝑎,𝑦,𝑧,𝑎2𝑞/𝑘𝑦𝑧 , 𝑘,− 𝑘, 𝑘𝑞 ,− 𝑘𝑞 ;𝑞;𝑎𝑞/𝑘

𝑎,− 𝑎,𝑎𝑞 /𝑦,𝑎𝑞 /𝑧,𝑘𝜌1𝜌2/𝑎,𝑎𝑞 / 𝑘,−𝑎𝑞 / 𝑘,𝑎 𝑞/𝑘,−𝑎 𝑞/𝑘

(1.2)

(𝑎𝑞 ,𝑎

2_𝑞/𝑘2_;𝑞) ∞ (𝑎𝑞 /𝑘,𝑎2_{𝑞/𝑘;𝑞)}

∞

×

𝑘+𝑎

𝑘 4

ɸ

3

𝑘,𝑘𝑦 /𝑎,𝑘𝑧 /𝑎,𝑎𝑞 /𝑦𝑧 ;𝑞;𝑎2/𝑘2 𝑎𝑞 /𝑦,𝑎𝑞 /𝑧,𝑘𝑦𝑧 /𝑎

=

10

ɸ

9

𝑎,𝑞 𝑎,−𝑞 𝑎,𝑦,𝑧,𝑎2_{𝑞/𝑘𝜌}

1𝜌2, 𝑘,− 𝑘, 𝑘𝑞 ,− 𝑘𝑞 ;𝑞;𝑎/𝑘

𝑎,− 𝑎,𝑎𝑞 /𝑦,𝑎𝑞 /𝑧,𝑘𝜌1𝜌2/𝑎,𝑎𝑞 / 𝑘,−𝑎𝑞 / 𝑘,𝑎 𝑞/𝑘,−𝑎 𝑞/𝑘

Here an attempt has been made to establish certain interesting transformation formulae for Basic

Hypergeometric functions by using some WP Bailey’s pairs.

2. Notation

Consider

𝑞 < 1,

where q is non-zero complex number, this condition ensures all the infinite products that we

use will converge. We will use the notation,

(2.1)

(𝛼; 𝑞)

𝑛

=

1 − 𝛼 1 − 𝛼𝑞 … … … 1 − 𝛼𝑞

𝑛−1

_{; 𝑛 > 0}

1 ; 𝑛 = 0,

(2.2)

(𝛼; 𝑞)

_∞

=

∞

1 − 𝛼𝑞

𝑟

_,

𝑟=0

(2.3)

(𝛼; 𝑞)

−𝑛

=

(−)𝑛_𝑞𝑛 (𝑛 +1)/2 𝛼𝑛_{[𝑞/𝑎 ; 𝑞]}_𝑛

,

(2)

Following the above notation, we define

(2.6)

r

ϕ

s𝑎1,𝑎_𝑏2₁,𝑎_,𝑏3₂………𝑎_{………𝑏}𝑟_𝑠 ;𝑞 ;𝑧

=

(𝑎1,𝑎2,𝑎3………𝑎𝑟 ;𝑞)𝑛𝑍𝑛 (𝑞,𝑏1,𝑏2………𝑏𝑠;𝑞)𝑛

∞

𝑛=0

Max

𝑞 , 𝑧 < 1 .

where

(𝑎

1

, 𝑎

2

… … … 𝑎

𝑟

; 𝑞)

𝑛

= (𝑎

1

; 𝑞)

𝑛

(𝑎

2

; 𝑞)

𝑛

… … 𝑎

𝑟

; 𝑞

𝑛

A basic series in which the product of each pair of numerator and denominator parameters is constant, is called a

well poised basic series. For example,

3

ϕ

2_{𝑎𝑞 /𝑏,𝑎𝑞 /𝑐}𝑎,𝑏,𝑐 ;𝑞 ;𝑧

, is said to be a series well-poised in

𝑎𝑞.

W.N.Bailey in 1944 stated a theorem which is simple but very use full. If

(2.7)

𝛽

𝑛

=

𝑛𝑟=0

𝛼

𝑟

𝑢

𝑛−𝑟

𝑣

𝑛+𝑟

(2.8)

𝛾

𝑛

=

∞𝑟=0

𝛿

𝑟+𝑛

𝑢

𝑛

𝑣

𝑟+2𝑛

where

𝛼

_𝑟

, 𝛽

_𝑟

, 𝑢

_𝑟

, 𝑎𝑛𝑑 𝑣

_𝑟

are functions of r alone , such that the series

𝛾

_𝑛

exists, then

(2.9)

∞_𝑛=0

𝛼

_𝑛

𝛾

_𝑛

=

∞_𝑛=0

𝛽

_𝑛

𝛿

_𝑛

We shall make use of the following known WP Bailey’s Pairs

(2.10)

𝛼

𝑛

(

𝑎, 𝑘; 𝑞)

=

_{(𝑞, 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌}(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2,;𝑞)𝑛

1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑘 𝑎

𝑛

,

&

𝛽

𝑛

(

𝑎, 𝑘; 𝑞)

=

[𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2 ;𝑞]𝑛 [𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞]𝑛

,

where m=

𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

[Mishra B. P.;6]

(2.11)

𝛼

_𝑛

(

𝑎, 𝑘; 𝑞)

=

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎

2_{𝑞/𝑏𝑐𝑚 ,𝜌}

1,𝜌2 ;𝑞)𝑛 (𝑞, 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑏𝑐𝑚 /𝑎,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑘 𝑎

𝑛

,

&

𝛽

_𝑛

(

𝑎, 𝑘; 𝑞)

=

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2 ;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

×

1−𝑚 𝑞2𝑟

1−𝑚

(𝑚 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝑎𝑞 /𝑏𝑐 ,𝜌1,𝜌2,𝑘𝑞𝑛,𝑞−𝑛;𝑞)𝑟𝑞𝑟 (𝑞,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑚𝑏𝑐 /𝑎,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝑚 𝑞1−𝑛/𝑘,𝑚 𝑞1+𝑛;𝑞)𝑟 𝑛

𝑟=0

,

where m=

𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

[Mishra B. P.;6]

We shall also make use of the following well known theorems.

(2.12) Theorem 1. If

{𝛼

𝑛

𝑎, 𝑘; 𝑞 , 𝛽

𝑛

𝑎, 𝑘; 𝑞 }

is a W.P. Bailey pair then

𝑎2𝑞

𝑘2 𝑛

∞

𝑛=0

𝛽

𝑛

𝑎, 𝑘; 𝑞 =

(𝑎𝑞 /𝑘, 𝑎 2_{𝑞/𝑘;𝑞)}

∞ (𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞)_∞

(𝑘;𝑞)2𝑛 (𝑎2_{𝑞/𝑘;𝑞)}_2𝑛

∞

𝑛=0

𝑎2𝑞 𝑘2

𝑛

𝛼

𝑛

𝑎, 𝑘; 𝑞

[Lauglin;4(1.7)]

(2.13) Theorem 2. If

{𝛼

𝑛

𝑎, 𝑘; 𝑞 , 𝛽

𝑛

𝑎, 𝑘; 𝑞 }

is a W.P. Bailey pair then

𝑎2 𝑘2

𝑛

∞

𝑛=0

𝛽

𝑛

𝑎, 𝑘; 𝑞 =

(𝑎𝑞 /𝑘, 𝑎2_{𝑞/𝑘;𝑞)} ∞ (𝑎𝑞 , 𝑎2_𝑞/𝑘2_{, ;𝑞)}_∞

𝑘

𝑘+𝑎

(𝑘;𝑞)2𝑛

(𝑎2_{𝑞/𝑘;𝑞)}_2𝑛

(1 + 𝑎𝑞

2𝑛

)

∞

𝑛=0

𝑎2 𝑘2

𝑛

𝛼

𝑛

𝑎, 𝑘; 𝑞

(3)

1−𝑘𝑞2𝑛 1−𝑘

∞

𝑛=0

(𝜌1,𝜌2;𝑞)𝑛 (𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛

𝑎𝑞 𝜌1𝜌2

𝑛

𝛽

_𝑛

𝑎, 𝑘; 𝑞

=

(𝑘𝑞 ,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)∞ (𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)∞

(𝜌1,𝜌2;𝑞)𝑛 (𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

∞

𝑛=0

𝑎𝑞 𝜌1𝜌2

𝑛

𝛼

𝑛

𝑎, 𝑘; 𝑞

[Lauglin 5; theorem 1]

3. Main Results

(3.1)

4

ɸ

3𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞;𝑎 2_𝑞/𝑘2

𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

=

[𝑎𝑞 /𝑘, 𝑎2_{𝑞/𝑘;𝑞]} ∞ [𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞]_∞

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2_𝑛

(𝑞, 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑎2𝑞/𝑘,𝑎2𝑞2/𝑘;𝑞2𝑛

∞

𝑛=0

𝑎𝑞 𝑘

𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(3.2)

4

ɸ

3𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞;𝑎 2_/𝑘2 𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

=

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞)∞

(𝑎𝑞 , 𝑎2_{𝑞/ 𝑘}2_;𝑞)_∞ 𝑘

𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2 _𝑛

∞

𝑛=0

𝑎 𝑘

𝑛

+

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞)∞ (𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞)_∞

𝑎𝑘

𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2𝑛

∞

𝑛=0

𝑎𝑞2 𝑘

𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(3.3)

6

ɸ

5 𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌_{𝑚𝑞 ,𝑘𝑞 /𝜌}1₁,𝑚𝑞 / 𝜌_{,𝑘𝑞 /𝜌}₂2_{,𝑎𝑞 /𝜌},𝜌1,𝜌2₁;𝑞;𝑎𝑞 /𝜌_{,𝑎𝑞 /𝜌}₂_,1𝜌2

− 𝑘

6

ɸ

5 𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 / 𝜌2,𝜌1,𝜌2;𝑞;𝑎𝑞 3_/𝜌

1𝜌2 𝑚𝑞 ,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,

=

[𝑘,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞]∞

[𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞]∞8

ɸ

7

𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2,𝜌1,𝜌2;𝑞;𝑘𝑞 /𝜌1𝜌2 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(3.4)

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑎2𝑞

𝑘2 𝑛

∞

𝑛=0 8

ɸ

7

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞

𝑛_,𝑞−𝑛_;𝑞;𝑞

𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑚𝑏𝑐 /𝑎,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝑚 𝑞1−𝑛/𝑘,𝑚 𝑞1+𝑛

]

𝑛

−𝑚

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞)𝑛 (𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑎2𝑞 𝑘2

𝑛

∞

𝑛=0 8

ɸ

7

[

𝑛_,𝑞−𝑛_;𝑞;𝑞3

]

𝑛

=

(𝑚 ,𝑎𝑞 /𝑘,𝑎2𝑞/𝑘;𝑞)∞

(𝑚𝑞 ,𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞) ∞

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌1,𝜌2 ;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2_𝑛 (𝑞, 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑏𝑐𝑚 /𝑎,𝑎𝑞 /𝜌1,𝑎𝑞/𝜌2;𝑞)𝑛 𝑎2𝑞/𝑘,𝑎2𝑞2/𝑘;𝑞2 𝑛

∞

𝑛=0

𝑎𝑞 𝑘

𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(3.5)

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌_{(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌}1,𝑚𝑞 /𝜌2;𝑞)𝑛

1,𝑎𝑞 /𝜌2;𝑞)𝑛

∞

𝑛=0

𝑎2 𝑘2

𝑛

8

ɸ

7

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞𝑛,𝑞−𝑛;𝑞;𝑞 𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑚𝑏𝑐 /𝑎,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝑚 𝑞1−𝑛/𝑘,𝑚 𝑞1+𝑛

]

𝑛

−𝑚

∞

𝑛=0

𝑎2 𝑘2

𝑛

8

ɸ

7

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞𝑛,𝑞−𝑛;𝑞;𝑞3 𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑚𝑏𝑐 /𝑎,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝑚 𝑞1−𝑛/𝑘,𝑚 𝑞1+𝑛

]

𝑛

=

(𝑚 ,𝑎𝑞/𝑘,𝑎2𝑞/𝑘;𝑞)∞

(𝑚𝑞 ,𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞) ∞

𝑘 𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌1,𝜌2 ;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2_𝑛 (𝑞, 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑏𝑐𝑚 /𝑎,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑎2𝑞/𝑘,𝑎2𝑞2/𝑘;𝑞2𝑛

∞

𝑛=0

𝑎 𝑘

𝑛

+

(𝑚𝑞 ,𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞) ∞

𝑎𝑘 𝑘+𝑎

∞

𝑛=0

𝑎𝑞2 𝑘

𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(3.6)

_{(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌}(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛

1,𝑎𝑞 /𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛 𝑎𝑞 𝜌1𝜌2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

−𝑘

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛 𝑎𝑞3 𝜌1𝜌2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛 𝑎𝑞 𝑛

(4)

+𝑚𝑘

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛 (𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛

𝑎𝑞3 𝜌1𝜌2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

=

(𝑚 ,𝑘,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)∞ (𝑚𝑞 ,𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)∞10

ɸ

9

𝑎 ,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝜌1,𝜌2,𝜌1,𝜌2,𝑎2𝑞/𝑏𝑐𝑚 ;𝑞;𝑘𝑞 /𝜌1𝜌2 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑏𝑐𝑚 /𝑎

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

4. Proof

(1) By using WP Bailey’s Pairs (2.10) in(2.12) theorem 1, we get

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2 ;𝑞)𝑛 (𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

∞

𝑛=0

𝑎2𝑞 𝑘2

𝑛

=

(𝑎𝑞 /𝑘,𝑎2𝑞/𝑘;𝑞)∞

(𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞)_∞

(𝑘;𝑞)2𝑛 (𝑎2_{𝑞/𝑘;𝑞)}_2𝑛

∞

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2,;𝑞)𝑛 (𝑞, 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑘 𝑎

𝑛

𝑎2𝑞

𝑘2 𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(4.1)

4

ɸ

3𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞;𝑎 2_𝑞/𝑘2

𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

=

[𝑎𝑞 /𝑘, 𝑎2_{𝑞/𝑘;𝑞]} ∞ [𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞]_∞

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2 𝑛

∞

𝑛=0

𝑎𝑞 𝑘

𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(2) By using WP Bailey’s Pairs (2.10) in (2.13) theorem 2, we get

∞

𝑛=0

𝑎2 𝑘2

𝑛

=

(𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞) ∞

𝑘 𝑘+𝑎

×

(𝑘;𝑞)2𝑛

(𝑎2_{𝑞/𝑘;𝑞)}_2𝑛

(1 + 𝑎𝑞

2𝑛

)

∞

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2,;𝑞)𝑛 (𝑞, 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑘 𝑎

𝑛

𝑎2

𝑘2 𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(4.2)

4

ɸ

3𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞;𝑎 2_/𝑘2

𝑚𝑞,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

=

(𝑎𝑞 , 𝑎2_{𝑞/ 𝑘}2_;𝑞)_∞ 𝑘

𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2 _𝑛

∞

𝑛=0

𝑎 𝑘

𝑛

+

(𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞)_∞ 𝑎𝑘

𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2_𝑛

∞

𝑛=0

𝑎𝑞2 𝑘

𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(3) By using WP Bailey’s Pairs (2.10) in (2.14) theorem 3, we get

1−𝑘𝑞2𝑛

1−𝑘

∞

𝑛=0

𝑎𝑞 𝜌1𝜌2

𝑛

=

(𝑘𝑞 ,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)∞ (𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)∞

∞

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 (𝑞, 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑘 𝑎

𝑛

𝑎𝑞

𝜌1𝜌2 𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(4.3)

6

ɸ

5𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌_{𝑚𝑞 ,𝑘𝑞 /𝜌}1₁,𝑚𝑞 / 𝜌_{,𝑘𝑞 /𝜌}₂2_{,𝑎𝑞 /𝜌},𝜌1,𝜌2₁;𝑞;𝑎𝑞 /𝜌_{,𝑎𝑞 /𝜌}₂_,1𝜌2

− 𝑘

6

ɸ

5𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 / 𝜌2,𝜌1,𝜌2;𝑞;𝑎𝑞 3_/𝜌

1𝜌2 𝑚𝑞 ,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,

=

_{(𝑎𝑞 ,𝑎𝑞 /𝜌}(𝑘,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)∞

1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)∞8

ɸ

7

𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2,𝜌1,𝜌2;𝑞;𝑘𝑞 /𝜌1𝜌2 𝑎,− 𝑎,𝑘𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

(5)

(4) By using WP Bailey’s Pairs (2.11) in (2.12) theorem 1, we get

1−𝑚 𝑞2𝑟

1−𝑚

𝑟=0

𝑎2𝑞 𝑘2

𝑛

∞

𝑛=0

=

(𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞)_∞

(𝑘;𝑞)2𝑛 (𝑎2_{𝑞/𝑘;𝑞)}_2𝑛

∞

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2_{𝑞/𝑏𝑐𝑚 ,𝜌}

𝑘 𝑎

𝑛

𝑎2𝑞

𝑘2 𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(4.4)

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑎2_𝑞

𝑘2 𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

−𝑚

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞)𝑛 (𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑎2𝑞 𝑘2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

=

(𝑚𝑞 ,𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞)_∞

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌1,𝜌2 ;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2_𝑛 (𝑞, 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑏𝑐𝑚 /𝑎,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑎2𝑞/𝑘,𝑎2𝑞2/𝑘;𝑞2 𝑛

∞

𝑛=0

𝑎𝑞 𝑘

𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(5) By using WP Bailey’s Pairs (2.11) in (2.13) theorem 2, we get

1−𝑚 𝑞2𝑟

1−𝑚

(𝑚 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝑎𝑞 /𝑏𝑐 ,𝜌1,𝜌2,𝑘𝑞𝑛,𝑞−𝑛;𝑞)𝑟𝑞𝑟 (𝑞,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑚𝑏𝑐 /𝑎,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝑚 𝑞1−𝑛/𝑘,𝑚 𝑞1+𝑛;𝑞)𝑟 𝑛 𝑟=0 ∞ 𝑛=0 𝑎2 𝑘2 𝑛

=

(𝑎𝑞 , 𝑎2_𝑞/𝑘2_;𝑞)_∞ 𝑘

𝑘+𝑎

(𝑘;𝑞)2𝑛

(𝑎2_{𝑞/𝑘;𝑞)}_2𝑛

(1 + 𝑎𝑞

2𝑛

)

∞

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌1,𝜌2 ;𝑞)𝑛 (𝑞, 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑏𝑐𝑚 /𝑎,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑘 𝑎 𝑛

𝑎2 𝑘2 𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(4.5)

∞

𝑛=0

𝑎2 𝑘2

𝑛

8

ɸ

7

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞 𝑛_,𝑞−𝑛_;𝑞;𝑞

]

𝑛

−𝑚

∞

𝑛=0

𝑎2 𝑘2

𝑛

8

ɸ

7

[

]

𝑛

=

(𝑚𝑞 ,𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞) ∞

𝑘 𝑘+𝑎

∞

𝑛=0

𝑎 𝑘

𝑛

+

(𝑚𝑞 ,𝑎𝑞 ,𝑎2_𝑞/𝑘2_;𝑞) ∞

𝑎𝑘 𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌1,𝜌2 ;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2 _𝑛 (𝑞, 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑏𝑐𝑚 /𝑎,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛 𝑎2𝑞/𝑘,𝑎2𝑞2/𝑘;𝑞2𝑛

∞

𝑛=0

𝑎𝑞2 𝑘

𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(6) By using WP Bailey’s Pairs (2.11) in (2.14) theorem 3, we get

1−𝑘𝑞2𝑛

1−𝑘

∞

𝑛=0

𝑎𝑞 𝜌1𝜌2

𝑛

×

1−𝑚 𝑞2𝑟

1−𝑚

𝑟=0

=

(𝑘𝑞 ,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2 ;𝑞)∞

(𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2 ;𝑞)∞

∞

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2_{𝑞/𝑏𝑐𝑚 ,𝜌}

𝑘 𝑎

𝑛

𝑎𝑞

𝜌1𝜌2 𝑛

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

(4.6)

(1−𝑘𝑞

2𝑛_{)(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌}

1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛 (𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛

𝑎𝑞 𝜌1𝜌2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

−𝑚

(1−𝑘𝑞2𝑛)(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛 𝑎𝑞 𝜌1𝜌2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

=

(𝑚 ,𝑘,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)∞ (𝑚𝑞 ,𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞/𝜌2;𝑞)∞10

ɸ

9

𝑎 ,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝜌1,𝜌2,𝜌1,𝜌2,𝑎2𝑞/𝑏𝑐𝑚 ;𝑞;𝑘𝑞 /𝜌1𝜌2 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑏𝑐𝑚 /𝑎

(6)

−𝑘

𝑛

∞

𝑛=0 8

ɸ

7

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞𝑛,𝑞−𝑛;𝑞;𝑞 𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑚𝑏𝑐 /𝑎,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝑚 𝑞1−𝑛/𝑘,𝑚 𝑞1+𝑛

]

𝑛

−𝑚

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2,𝜌1,𝜌2;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛 𝑎𝑞 𝜌1𝜌2

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

+𝑚𝑘

𝑛

∞

𝑛=0 8

ɸ

7

[

]

𝑛

=

(𝑚 ,𝑘,𝑘𝑞 /𝜌1𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)∞ (𝑚𝑞 ,𝑎𝑞 ,𝑎𝑞 /𝜌1𝜌2,𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)∞10

ɸ

9

𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝜌1,𝜌2,𝜌1,𝜌2,𝑎2𝑞/𝑏𝑐𝑚 ;𝑞;𝑘𝑞 /𝜌1𝜌2 𝑎,− 𝑎,𝑎𝑞 /𝑏,𝑎𝑞 /𝑐,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2,𝑏𝑐𝑚 /𝑎

where

𝑚 = 𝑘𝜌

₁

𝜌

₂

/𝑎𝑞

References

[1] Andrews, G.E. and Berndt, B.C., Ramanujans Lost Notebook, Part I, Springer, 2005.

[2] Denis, R. Y. : Certaian Transformation and Summation Formulae for q-Series, Italian J. of Pure and Applied Mathematics-N. 27, pp 179-190, 2010.

[3] Gasper, G. and Rahaman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1991.

[4] Laughlin :J.M., Some identities between basic hypergeometric series deriving from a new Bailey type transformation, The J.of mathematical Analysis and Applications, 345(2), pp 670-677, 2008.

[5] Laughlin :J.M. and Zimmer, P., Some applications of a Bailey-type transformations International Mathematical Forum, No. 61,pp 3007-3022, 2010.

[6] Mishra, B. P. ,On summation and transformation formulae for basic hypergeometric series, J.of Ramanujan Society of Math.and Math. Sc.Vol.2, No.1, pp.53-72, 2013.

[7] Singh, S.N., Baiey transform, WP-Bailey pairs and q-series transformations, J.of Ramanujan Society of Math. and Math. Sc.Vol.2, No.1, pp.85-92, 2013.