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

(1)

## Baley’s Pairs

#1

2

#1

*2

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

𝑛

𝑛−1

𝑟

𝑟=0

−𝑛

### =

(−)𝑛𝑞𝑛 (𝑛 +1)/2 𝛼𝑛[𝑞/𝑎 ; 𝑞]𝑛

(2)

r

### ϕ

s𝑎1,𝑎𝑏21,𝑎,𝑏32………𝑎………𝑏𝑟𝑠 ;𝑞 ;𝑧

### =

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

𝑛=0

1

2

𝑟

𝑛

1

𝑛

2

𝑛

𝑟

𝑛

3

### ϕ

2𝑎𝑞 /𝑏,𝑎𝑞 /𝑐𝑎,𝑏,𝑐 ;𝑞 ;𝑧

𝑛

𝑛𝑟=0

𝑟

𝑛−𝑟

𝑛+𝑟

𝑛

𝑟=0

𝑟+𝑛

𝑛

𝑟+2𝑛

𝑟

𝑟

𝑟

𝑟

𝑛

𝑛=0

𝑛

𝑛

𝑛=0

𝑛

𝑛

𝑛

### =

(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

𝑛

### =

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

1

2

𝑛

### =

(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

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

1

2

𝑛

𝑛

𝑎2𝑞

𝑘2 𝑛

𝑛=0

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

𝑛

𝑛

𝑛

𝑎2 𝑘2

𝑛

𝑛=0

𝑛

### 𝑎, 𝑘; 𝑞 =

(𝑎𝑞 /𝑘, 𝑎2𝑞/𝑘;𝑞) (𝑎𝑞 , 𝑎2𝑞/𝑘2, ;𝑞)

𝑘

𝑘+𝑎

(𝑘;𝑞)2𝑛

(𝑎2𝑞/𝑘;𝑞)2𝑛

2𝑛

𝑛=0

𝑎2 𝑘2

𝑛

𝑛

### 𝑎, 𝑘; 𝑞

(3)

1−𝑘𝑞2𝑛 1−𝑘

𝑛=0

(𝜌1,𝜌2;𝑞)𝑛 (𝑘𝑞 /𝜌1,𝑘𝑞 /𝜌2;𝑞)𝑛

𝑎𝑞 𝜌1𝜌2

𝑛

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

𝑎𝑞 𝜌1𝜌2

𝑛

𝑛

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

1

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

1

2

6

### ɸ

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

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

1

2

### (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

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

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

1

2

### (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

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

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

1

2

### (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

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

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

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

1

2

### (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

1

2

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

1

2

### (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

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

1

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

1

2

### (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

1

2

6

### ɸ

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

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

1

2

### (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

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

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

1

2

### (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

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

1

2

### (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

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

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

1

2

### (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

1

2

### (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

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

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

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

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

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

1

2

### 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.

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