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

Download (0)

Full text

(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 2𝑞/𝑘2

𝑎𝑞 /𝑦,𝑎𝑞 /𝑧,𝑘𝑦𝑧 /𝑎

=

(𝑎𝑞 /𝑘,𝑎2𝑞/𝑘;𝑞) (𝑎𝑞 ,𝑎2𝑞/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

2𝑞/𝑘2;𝑞) (𝑎𝑞 /𝑘,𝑎2𝑞/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,𝑎2𝑞/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

=

(−)𝑛𝑞𝑛 (𝑛 +1)/2 𝛼𝑛[𝑞/𝑎 ; 𝑞]𝑛

,

(2)

Following the above notation, we define

(2.6)

r

ϕ

s𝑎1,𝑎𝑏21,𝑎,𝑏32………𝑎………𝑏𝑟𝑠 ;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

2𝑎𝑞 /𝑏,𝑎𝑞 /𝑐𝑎,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

2𝑞/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 2𝑞/𝑘;𝑞)

(𝑎𝑞 , 𝑎2𝑞/𝑘2 ;𝑞)

(𝑘;𝑞)2𝑛 (𝑎2𝑞/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, 𝑎2𝑞/𝑘;𝑞) (𝑎𝑞 , 𝑎2𝑞/𝑘2, ;𝑞)

𝑘

𝑘+𝑎

(𝑘;𝑞)2𝑛

(𝑎2𝑞/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 2𝑞/𝑘2

𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

=

[𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞] [𝑎𝑞 , 𝑎2𝑞/𝑘2 ;𝑞]

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2𝑛

(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 , 𝑎2𝑞/ 𝑘2 ;𝑞) 𝑘

𝑘+𝑎

(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

𝑎 𝑘

𝑛

+

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞) (𝑎𝑞 , 𝑎2𝑞/𝑘2 ;𝑞)

𝑎𝑘

𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2𝑛

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

𝑛=0

𝑎𝑞2 𝑘

𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(3.3)

6

ɸ

5 𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌𝑚𝑞 ,𝑘𝑞 /𝜌11,𝑚𝑞 / 𝜌,𝑘𝑞 /𝜌22,𝑎𝑞 /𝜌,𝜌1,𝜌21;𝑞;𝑎𝑞 /𝜌,𝑎𝑞 /𝜌2,1𝜌2

− 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

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞

𝑛,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 ,𝑎2𝑞/𝑘2 ;𝑞)

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

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,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;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 ,𝑎2𝑞/𝑘2 ;𝑞)

𝑘 𝑘+𝑎

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

+

(𝑚 ,𝑎𝑞 /𝑘,𝑎2𝑞/𝑘;𝑞)

(𝑚𝑞 ,𝑎𝑞 ,𝑎2𝑞/𝑘2;𝑞)

𝑎𝑘 𝑘+𝑎

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

(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

[

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

−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

[

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

(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

[

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 ,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 , 𝑎2𝑞/𝑘2 ;𝑞)

(𝑘;𝑞)2𝑛 (𝑎2𝑞/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 2𝑞/𝑘2

𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2

=

[𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞] [𝑎𝑞 , 𝑎2𝑞/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

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

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

𝑛=0

𝑎2 𝑘2

𝑛

=

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞)

(𝑎𝑞 ,𝑎2𝑞/𝑘2 ;𝑞)

𝑘 𝑘+𝑎

×

(𝑘;𝑞)2𝑛

(𝑎2𝑞/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 /𝑘, 𝑎2𝑞/𝑘;𝑞)

(𝑎𝑞 , 𝑎2𝑞/ 𝑘2 ;𝑞) 𝑘

𝑘+𝑎

(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

𝑎 𝑘

𝑛

+

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞)

(𝑎𝑞 , 𝑎2𝑞/𝑘2 ;𝑞) 𝑎𝑘

𝑘+𝑎

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑎/𝑘,𝜌1,𝜌2;𝑞)𝑛 𝑘,𝑘𝑞 ;𝑞2𝑛

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

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

(𝜌1,𝜌2;𝑞)𝑛 (𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛

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

𝑎𝑞 𝜌1𝜌2

𝑛

=

(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

(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 /𝜌𝑚𝑞 ,𝑘𝑞 /𝜌11,𝑚𝑞 / 𝜌,𝑘𝑞 /𝜌22,𝑎𝑞 /𝜌,𝜌1,𝜌21;𝑞;𝑎𝑞 /𝜌,𝑎𝑞 /𝜌2,1𝜌2

− 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

(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

𝑎2𝑞 𝑘2

𝑛

𝑛=0

=

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞)

(𝑎𝑞 , 𝑎2𝑞/𝑘2 ;𝑞)

(𝑘;𝑞)2𝑛 (𝑎2𝑞/𝑘;𝑞)2𝑛

𝑛=0

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌

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

𝑘 𝑎

𝑛

𝑎2𝑞

𝑘2 𝑛

where

𝑚 = 𝑘𝜌

1

𝜌

2

/𝑎𝑞

(4.4)

(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

[

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 ,𝑎2𝑞/𝑘2 ;𝑞)

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

(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 𝑛=0 𝑎2 𝑘2 𝑛

=

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞)

(𝑎𝑞 , 𝑎2𝑞/𝑘2 ;𝑞) 𝑘

𝑘+𝑎

(𝑘;𝑞)2𝑛

(𝑎2𝑞/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)

(𝑘,𝑘/𝑚 ,𝑚𝑞 /𝜌1,𝑚𝑞 /𝜌2;𝑞)𝑛

(𝑞,𝑚𝑞 ,𝑎𝑞 /𝜌1,𝑎𝑞 /𝜌2;𝑞)𝑛

𝑛=0

𝑎2 𝑘2

𝑛

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

𝑛=0

𝑎2 𝑘2

𝑛

8

ɸ

7

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞

𝑛,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 ,𝑎2𝑞/𝑘2 ;𝑞)

𝑘 𝑘+𝑎

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

+

(𝑚 ,𝑎𝑞 /𝑘,𝑎2𝑞/𝑘;𝑞)

(𝑚𝑞 ,𝑎𝑞 ,𝑎2𝑞/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

(𝜌1,𝜌2;𝑞)𝑛 (𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛

𝑎𝑞 𝜌1𝜌2

𝑛

×

(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

=

(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

(𝑎,𝑞 𝑎,−𝑞 𝑎,𝑏,𝑐,𝑎2𝑞/𝑏𝑐𝑚 ,𝜌

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

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

[

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

(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

[

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 ,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

(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

[

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

[

𝑚 ,𝑎𝑞 /𝑏𝑐 ,𝑚𝑏 /𝑎,𝑚𝑐 /𝑎,𝜌1,𝜌2,𝑘𝑞

𝑛,𝑞−𝑛;𝑞;𝑞3

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

]

𝑛

+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

[

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 ,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.

Figure

Updating...

References

Related subjects :