Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

369 https://internationalpubls.com 

The T-Generalized P - K Wright Function and its Properties 

 

1Arvind Maharshi, 2Anita, 3Krishnapal Singh Sisodia 
1,2School of Engineering and Technology, Mody University of Science and Technology, Lakshmangarh, Sikar,Rajasthan, 

India-332311 

3School of Liberal Arts and Sciences, Mody University of Science and Technology, Lakshmangarh, Sikar, Rajasthan, 

India-332311 

Email: 1maharshiarvind1973@gmail.com, 2anita.neetu89@yahoo.com, 3sisodiakps@gmail.com 

 

Article History: 

Received: 25-10-2024 

Revised:10-11-2024 

Accepted:18-12-2024 

 

 

 

Abstract:  

In this paper, we introduce t-Generalized p-k Wright function (_r^(t,p))Ψ_s^k  and 

discuss about its convergence condition. We obtain functional relation between 

t-Generalized p-k Wright function, Generalized p-k Wright function and generalized 

Wright function and some special cases have also been discussed. We obtain integral 

representation of t-Generalized p-k Wright function and discussed its properties. 

Keywords: The t-Generalized p- k Wright function, Generalized p-k Wright function, 

Generalized k-Wright function, Generalized Wright function, two parameter 

Pochhammer symbol, two parameter Gamma function. 

MSC(2011): 26A33, 33B15, 33C20, 33E12, 33B10. 

 

1. Introduction 

The two parameter Pochhammer symbol is recently introduce by [8,9], equation 13, in the form of the 

following definitions: 

1.1. Definition 

Let {0},; − +RpkCx  and ,0,>)( NnxRe   the p - k Pochhammer symbol  (two parameter ), 

knp x ,)(  is given by  

 ).1)(().........2)()((=)( , pn
k

xp
p

k

xp
p

k

xp

k

xp
x knp −+++  (1) 

 and the two parameter Gamma function is given by [6]. 

1.2. Definition 

For {0},;/ − +− RpkkZCx  and ,0,>)( NnxRe   the p - k Gamma function (two parameter), 

)(xkp  is given as,  

 .
)(

)(!
lim

1
=)(

1,

1

knp

k

x

n

n
kp

x

nppn

k
x

+

+

→

  (2) 

 or  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

370 https://internationalpubls.com 

 .
)(

)(!
lim

1
=)(

,

1
1

knp

k

x

n

n
kp

x

nppn

k
x

−
+

→

  (3) 

 The integral representation of p - k Gamma Function is given by,  

 .=)( 1

0
dttex xp

kt

kp

−
−

  (4) 

Also it is easy to prove following results, 

 ).(=)()(=)(
k

x

k

p
x

k

p
x

k

x

k
k

x

kp   (5) 

 .)()(=)()(=)( ,, n

n

kn

n

knp
k

x
px

k

p
x  (6) 

 .
)(

)(
=)( ,

x

nkx
x

kp

kp

knp


+
 (7) 

 ).(=)( x
k

xp
kx kpkp +  (8) 

 .

)(

1
=)()(

k

x
sin

xk
xx kpkp 


−  (9) 

 .

)(

=)()(
2

k

x
sin

k

p
xkx kpkp 


−  (10) 

 .)()(=)( ,,1, knpknpknp kxxxn −−−  (11) 

 .)()(=)( ,,, knpkjpkjnp jkxxx ++  (12) 

2. t-Generalized p-k Wright Function 

t-Generalized p-k Wright function is denoted by 𝛹𝑠
𝑘

𝑟
𝑡,𝑝

 and defined as 

                  𝛹𝑠
𝑘

𝑟
𝑡,𝑝 (𝑧) = ∑

∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0       (13) 

where {0}, − +Rkp  𝑡 ∈ 𝑵𝟎;, Cz , )1,2,...,=;1,2,...,=0;,(, sjriR jiji     

and ,\)(),( −++ kZCnbna jjii    

We use following notations for describing convergence condition, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

371 https://internationalpubls.com 

;||||=);()(=
1=1=1=1=

k

j
j

s

j

k

i
i

r

i

i
r

i

j
s

j kkkk








−

−  

.
2

)()(=
1=1=

sr

k

a

k

b
i

r

i

j
s

j

−
+−  

Theorem 1. For {0}, − +Rkp ; 𝑡 ∈ 𝑵𝟎 Cz , )1,2,...,=;1,2,...,=0;,(, sjriR jiji     

     and ,\)(),( −++ kZCnbna jjii   

(a) If 1> −  then series (13) is absolutely convergent for all Cz  and t-Generalized p-k       

   Wright function  𝛹𝑠
𝑘

𝑟
𝑡,𝑝 (𝑧) is an entire function of z. 

(b) If 1= −  then series (13) is absolutely convergent for all |<| z   

    and of |=| z , .
2

1
>)(Re  

Proof:  Above theorem can prove easily by using results of Diaz and Pariguan [1], Kilbas [10],     

         K.S. Gehlot [5,6]. 

3. Special cases 

For some particular values of the parameters, we can obtain certain Wright function and Mittag-Leffler 

function defined earlier. 

(i)For 𝑡 = 0 equation (13), reduces in generalized p-k Wright function [4 ] as 

                 𝛹𝑠
𝑘

𝑟
𝑝 (𝑧) = ∑

∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖𝑛)   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1 𝑛!

∞
𝑛=0  . 

(ii) For 𝑡 = 0 , kp =  equation (13), reduces in generalized k-Wright Function [5] as 

 𝛹𝑠
𝑘

𝑟
𝑘 (𝑧) = ∑

∏ 𝛤𝑘𝑘 (𝑎𝑖+𝛼𝑖𝑛)   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑘 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1 𝑛!

∞
𝑛=0  = 𝛹𝑠

𝑘(𝑧)𝑟  

(iii) For 𝑝 = 𝑘 = 1 equation (13), reduces in t-Generalized Wright Function as, 

                𝛹𝑠
1

𝑟
𝑡,1 (𝑧) = ∑

∏ 𝛤11 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤11 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  = 𝛹𝑠 (𝑧)𝑟

𝑡  

(iv) For kp =  equation (13), reduces in t-generalized k-Wright Function as, 

 𝛹𝑠
𝑘

𝑟
𝑡,𝑘 (𝑧) = ∑

∏ 𝛤𝑘𝑘 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑘 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1

(𝑛+𝑡)!

∞
𝑛=0   

(v) For 𝑡 = 0 kp =  and 1=k  equation (13), reduces in generalized Wright Function 10] as 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

372 https://internationalpubls.com 

                  ).(=
!

)(

)(

=)(

11

1=

11

1=

0=

11 z
n

z

nb

na

z sr

n

jj

s

j

ii

r

i

n

sr 







+

+







 

(vi) For 𝑡 = 0  , 1=r , 2=s ; =1a , qk=1  where Nq  (0,1) ; =1b ,  =1
; =2b , 

0=2 , in equation (13) it reduces in p-k Mittag-Leffler function[7,8] as 

    ).(=
!)()(

)(
=)( ,

,,

0=

21 zE
n

z

n

qkn
z q

kp

n

kpkp

kp

n

kp 







+

+



 

(vii) For 𝑡 = 𝑗  , 1=r , 2=s ; =1a , qk=1  where Nq  (0,1) ; =1b ,  =1
; =2b , 

0=2  in equation (13) it reduces in j-generalized p-k Mittag-Leffler function as, 

                    𝛹2
𝑘

1

𝑗,𝑝 (𝑧) = ∑
𝛤𝑘 (𝑝 𝛾+𝑞𝑘(𝑛+𝑗)           𝑧𝑛

𝛤𝑘 (𝑝 𝛽+𝛼𝑛) 𝛤𝑘𝑝 (𝛾)(𝑛+𝑗)!

∞
𝑛=0    

    

Using equation (7) it can be written as 

                    𝛹2
𝑘

1

𝑗,𝑝 (𝑧) = ∑
(𝛾)(𝑛+𝑗)𝑞,𝑘𝑝            𝑧𝑛

𝛤𝑘 (𝑝 𝛽+𝛼𝑛)    (𝑛+𝑗)!

∞
𝑛=0  = 𝐸𝑘,𝛼,𝛽

𝛾,𝑞
𝑝
𝑗

(𝑧)  . 

Now it follows all particular cases of  j-generalized p-k Mittag-Leffler function given by K.S Gehlot 

and Anjana Bhandhari . 

(viii) For 𝑡 = 0, 1=r , 2=s ; and kp = , =1a , qk=1  where Nq  (0,1) ; =1b ,  =1
; 

=2b , 0=2 , in equation (13) it reduces in generalized k- Mittag-Leffler function[3] as

 ).(=
!)()(

)(
=)( ,

,,

0=

21 zGE
n

z

n

qkn
z q

k

n

kkkk

kk

n

kk 







+

+



 

(ix) For t=0, 1=r , 2=s  and kp = , =1a , k=1 ; =1b ,  =1
; =2b , 0=2 , in equation 

(13) it reduces in k-Mittag-Leffler function[2] as 

 ).(=
!)()(

)(
=)( ,,

0=

21 zE
n

z

n

kn
z k

n

kkkk

kk

n

kk 







+

+



 

(x) For 𝑡 = 0, 1=r , 2=s ; and 1== kp , =1a , 1=1 ; =1b ,  =1
; =2b , 0=2 , in 

equation (13) it reduces in Mittag-Leffler function[12] as 

 ).(=
!)()(

)(
=)( ,

1111

11

0=

1

2

1

1 zE
n

z

n

n
z

n

n









+

+



 

(xi) For t=0 1=r , 2=s  and 1== kp , 1=1a , 1=1 ; =1b ,  =1
; 1=2b , 0=2 , in 

equation (13) it reduces in Mittag-Leffler Function[13] as 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

373 https://internationalpubls.com 

 ).(=
!(1))(

)(1
=)( ,

11

1

0=

1

2

1

1 zE
n

z

n

n
z

n

n





+

+



 

 

(xii) For t=0 , 1=r , 2=s  and 1== kp , 1=1a , 1=1 ; 1=1b ,  =1
; 1=2b , 0=2 , in 

equation (13) it reduces in Mittag-Leffler function [11] as 

 ).(=
!(1))(1

)(1
=)(

11

1

0=

1

2

1

1 zE
n

z

n

n
z

n

n





+

+



 

4. Properties of Generalized p-k Wright Function 

We evaluate the functional relation between t-Generalized p-k Wright function,Generalized p-k 

Wright function and Generalized Wright function and we obtain the recurrence relations of  

t-Generalized p-k Wright function. 

Theorem 2. The functional relation between t-Generalized p-k Wright function and Generalized p-k 

Wright function, is given by,  

𝛹𝑠
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝 =

                   
𝑝

∑ (
𝑎𝑖
𝑘

+
𝛼𝑖𝑡

𝑘
)−∑ (

𝑏𝑗
𝑘

)𝑠
𝑗=1

𝑟
𝑖=1

𝑘𝑟−𝑠 𝛹𝑠

[
 
 
 
 (

𝑎𝑖

𝑘
,
𝛼𝑖𝑡

𝑘
)1,𝑟;

𝑧𝑝
∑ (

𝛼𝑖
𝑘

)𝑟
𝑖=1 −∑ (

𝛽𝑗

𝑘
)𝑠

𝑗=1

(
𝑏𝑗

𝑘
,
𝛽𝑗

𝑘
)1,𝑠; ]

 
 
 
 

𝑟
𝑡                    (14) 

 Or the counter part, 

𝛹𝑠 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠;

]𝑟
𝑡 =

𝑝
∑ (𝑏𝑗)−∑ (𝑎𝑖+𝛼𝑖𝑡)

𝑟
𝑖=1

𝑠
𝑗=1

𝑘𝑠−𝑟
𝛹𝑠

𝑘 [

(𝑘𝑎𝑖, 𝑘𝛼𝑖𝑡)1,𝑟;

𝑧𝑝∑ 𝛽𝑗
𝑠
𝑗=1 −∑ 𝛼𝑖

𝑟
𝑖=1

(𝑘𝑏𝑗 , 𝑘𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝

                           

(15)                                                                 (15) 

Proof: Consider the right hand side of (14), and using equation (13), we have, 

 𝐴 ≡ 𝛹𝑠
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝 = ∑

∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

Using equation (5), we have, 𝐴 ≡ ∑
∏

𝑝

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘

𝑘
  𝛤(

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘
)  𝑧𝑛𝑟

𝑖=1

∏
𝑝

𝑏𝑗+𝛽𝑗𝑛

𝑘

𝑘
  𝛤(

𝑏𝑗+𝛽𝑗 𝑛

𝑘
)  𝑠

𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

 𝐴 ≡  
𝑝

∑ 𝑎𝑖+𝛼𝑖𝑡−∑ 𝑏𝑗
𝑠
𝑗=1

𝑟
𝑖=1

𝑘

𝑘𝑟−𝑠    ∑
∏ 𝛤(

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘
)  (𝑧𝑝

∑ 𝛼𝑖  
𝑟
𝑖=1 −∑ 𝛽𝑗

𝑠
𝑗=1

𝑘   )𝑛𝑟
𝑖=1

∏  𝛤(
𝑏𝑗+𝛽𝑗 𝑛

𝑘
)  𝑠

𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

374 https://internationalpubls.com 

 𝐴 ≡
𝑝

∑ (
𝑎𝑖
𝑘

+
𝛼𝑖𝑡

𝑘
)−∑ (

𝑏𝑗
𝑘

)𝑠
𝑗=1

𝑟
𝑖=1

𝑘𝑟−𝑠
    𝛹𝑠

[
 
 
 
 (

𝑎𝑖

𝑘
,
𝛼𝑖𝑡

𝑘
)1,𝑟;

𝑧𝑝
∑ (

𝛼𝑖
𝑘

)𝑟
𝑖=1 −∑ (

𝛽𝑗

𝑘
)𝑠

𝑗=1

(
𝑏𝑗

𝑘
,
𝛽𝑗

𝑘
)1,𝑠; ]

 
 
 
 

𝑟
𝑡  

Similarly we can Prove counterpart, (15). 

Theorem 3. Let {0}, − +Rkp ; 𝑡 ∈ 𝑵𝟎 Cz , )1,2,...,=;1,2,...,=0;,(, sjriR jiji     

and ,\)(),( −++ kZCnbna jjii   then,  

𝛹𝑠+1
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠, (𝑏, 𝛽);

]𝑟
𝑡,𝑝 =

𝑏𝑝

𝑘
 𝛹𝑠+1

𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠, (𝑏 + 𝑘, 𝛽);

]𝑟
𝑡,𝑝 +

 
𝛽𝑧𝑝

𝑘
   

𝑑

𝑑𝑧
𝛹𝑠+1

𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠, (𝑏 + 𝑘, 𝛽);

]𝑟
𝑡,𝑝

  (16) 

Proof: Consider the right hand side of (16) and using the definition of t-Generalized p-k Wright 

function (13), we have,  

 𝐵 ≡
𝑝𝑏

𝑘
∑

∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛) 𝛤𝑘𝑝 (𝑏+𝑘+𝛽𝑛)  𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0    

                                         +
𝛽𝑧𝑝

𝑘
 

𝑑

𝑑𝑧
∑

∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛) 𝛤𝑘𝑝 (𝑏+𝑘+𝛽𝑛)  𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0   

Using equation (8), we get, 

 𝐵 ≡ ∑
𝑝

𝑘

∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖𝑡) (𝑏+𝛽𝑛)   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛) 𝛤𝑘𝑝 (𝑏+𝑘+𝛽𝑛)  𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

                   𝐵 ≡ ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟

𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛) 𝛤𝑘𝑝 (𝑏+𝛽𝑛)  𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

this immediately leads to,  𝛹𝑠+1
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠, (𝑏, 𝛽);

]𝑟
𝑡,𝑝

 

Theorem 4. Let {0}, − +Rkp ; 𝑡 ∈ 𝑵𝟎 Cz , )1,2,...,=;1,2,...,=0;,(, sjriR jiji     

and ,\)(),( −++ kZCnbna jjii   then,  

𝛹𝑠+1
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟 , (𝑎 + 𝑘, 𝛼𝑘𝑡);

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠, (𝑎 + 𝑘, 0);

]𝑟+1
𝑡,𝑝 − 𝛹𝑠+1

𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟 , (𝑎, 𝛼𝑡𝑘);

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠, (𝑎, 0);

]𝑟+1
𝑡,𝑝

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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375 https://internationalpubls.com 

= 𝛼𝑧 (𝑎 + 𝑘)𝛼−1,𝑘
   𝑝 𝛹𝑠+1

𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟 , (𝑎 + 𝛼𝑘, 𝛼𝑘𝑡);

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠, (𝑎 + 𝛼𝑘, 0);

]𝑟+1
𝑡,𝑝                (17) 

Proof: Consider the right hand side of (17) and using the definition of t-Generalized p-k Wright 

function (13),  

we have, 

𝛹𝑠+1
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟 , (𝑎 + 𝑘, 𝛼𝑘𝑡);

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠, (𝑎 + 𝑘, 0);

]𝑟+1
𝑡,𝑝

  

− 𝛹𝑠+1
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟 , (𝑎, 𝛼𝑡𝑘);

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠, (𝑎, 0);

]𝑟+1
𝑡,𝑝

 

         ≡ ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))  𝛤𝑘𝑝 (𝑎+𝑘+𝛼𝑘(𝑛+𝑡)) 𝑧𝑛𝑟

𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛)  𝛤𝑘𝑝 (𝑎+𝑘) 𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

Using equation (11) and (12), we get, 

 ≡ ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟

𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0 ( (𝑎 + 𝑘)𝛼(𝑛+𝑡),𝑘 − (𝑎)𝛼(𝑛+𝑡),𝑘)𝑝𝑝  

         ≡ ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡))   𝑧𝑛𝑟

𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛)   𝑠
𝑗=1

(𝑛+𝑡)!

∞
𝑛=0 ((𝑛 + 𝑡)𝛼 (𝑎 + 𝑘)𝛼(𝑛+𝑡)−1,𝑘)𝑝  

         ≡ ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+1+𝑡))   𝑧𝑛+1𝑟

𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗(𝑛+1))   𝑠
𝑗=1

(𝑛+1+𝑡)!

∞
𝑛=0 ((𝑛 + 1 + 𝑡)𝛼 (𝑎 + 𝑘)𝛼(𝑛+1+𝑡)−1,𝑘)𝑝  

         ≡ ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡)+𝛼𝑖)   𝑧

𝑛+1𝑟
𝑖=1 𝛼(𝑛+1+𝑡)

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛+𝛽𝑗)   
𝑠
𝑗=1

(𝑛+1+𝑡)!

∞
𝑛=0 ( (𝑎 + 𝑘)𝛼−1,𝑘  (𝑎 + 𝑘 + (𝛼 − 1)𝑘𝑝 )(𝑛+𝑡)𝛼,𝑘)𝑝  

 ≡ 𝛼𝑧 (𝑎 + 𝑘)𝛼−1,𝑘    ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡)+𝛼𝑖)   𝑧

𝑛 (𝑛+1+𝑡)𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗𝑛+𝛽𝑗)   
𝑠
𝑗=1

(𝑛+1+𝑡)!

∞
𝑛=0    (𝑎 + 𝑘)𝛼(𝑛+𝑡),𝑘)𝑝𝑝  

Using equation (13), we have,  

= 𝛼𝑧 (𝑎 + 𝑘)𝑝 𝛼−1,𝑘     𝛹𝑠+1
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟 , (𝑎 + 𝛼𝑘, 𝛼𝑘𝑡);

𝑧
(𝑏𝑗 + 𝛽𝑗 , 𝛽𝑗)1,𝑠, (𝑎 + 𝛼𝑘, 0);

]𝑟+1
𝑡,𝑝

 

5. Integral Representation of Generalized p-k Wright Function 

Theorem 5. Let {0}, − +Rkp ; 𝑡 ∈ 𝑵𝟎 Cz , )1,2,...,=;1,2,...,=0;,(, sjriR jiji    

and ,\)(),( −++ kZCnbna jjii   then, 

 𝛹𝑠
𝑘 [

(𝑎𝑖, 𝑘𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗, 𝑘𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝

 



Communications on Applied Nonlinear Analysis 

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Vol 32 No. 7s (2025) 

 

376 https://internationalpubls.com 

 =
∏ 𝛤𝑘 (𝑎𝑖)𝑝

𝑟
𝑖=1

∏ 𝛤𝑘 (𝑏𝑖)𝑝
𝑠
𝑗=1

   ∏ ∏
1

𝐵(𝜇𝑙,𝜗𝑚−𝜇𝑙)
∫ 𝑈𝜇𝑙−1(1 − 𝑈)𝜗𝑚−𝜇𝑙−1 𝑒

(𝛼𝑖𝑝)𝛼𝑖𝑈𝑧

(𝑝𝛽𝑗)
𝛽𝑗

𝑑𝑈 
1

0

𝛽𝑗

𝑚=1

𝛼𝑖
𝑙=1       (18)       

                                          

Here 𝜇𝑙 = 
𝑎𝑖
𝑘

+𝑡𝛼𝑖+𝑙−1

𝛼𝑖
 𝑎𝑛𝑑 𝜗𝑚 = 

𝑏𝑗

𝑘
+𝑚−1

𝛽𝑗
  . 

Proof: Using the definition of t-Generalized p-k Wright function 

𝛹𝑠
𝑘 [

(𝑎𝑖, 𝑘𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗 , 𝑘𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝 = ∑

∏ 𝛤𝑘𝑝 (𝑎𝑖 + 𝑘𝛼𝑖(𝑛 + 𝑡))   𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗 + 𝑘𝛽𝑗𝑛)   𝑠
𝑗=1 (𝑛 + 𝑡)!

∞

𝑛=0

 

 = ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖)  (𝑎𝑖)𝑡𝛼𝑖,𝑘

     𝑝   (𝑎𝑖+𝑡𝛼𝑖𝑘)𝑛𝛼𝑖,𝑘        𝑝 𝑧𝑛𝑟
𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗)    (𝑏𝑗)𝑛𝛽𝑗,𝑘𝑝          𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

               = ∑ 𝐷  
(𝑎𝑖)𝑡𝛼𝑖,𝑘 

 
𝑝   𝛤𝑘 (𝑎𝑖)          𝑧

𝑛
𝑝   

𝛤𝑘 (𝑏𝑗)𝑝         (𝑛+𝑡)!

∞
𝑛=0  (19) 

 Where, 𝐷 ≡  
∏ (𝑎𝑖+𝛼𝑖𝑡𝑘)𝑛𝛼𝑖,𝑘𝑝

𝑟
𝑖=1

∏ (𝑏𝑗)𝑛𝛽𝑗,𝑘𝑝
𝑠
𝑗=1

 . 

Using equation (6) we have, 𝐷 ≡  
∏ 𝑝

𝑛(𝛼𝑖−𝛽𝑗)𝑟
𝑖=1  (

𝑎𝑖+𝑡𝛼𝑖𝑘

𝑘
)𝑛𝛼𝑖

∏ (
𝑏𝑗

𝑘
)𝑛𝛽𝑗

𝑠
𝑗=1

 

Also using the relation given by equation equation (7) we have, 

 𝐷 ≡  
∏ 𝑝

𝑛(𝛼𝑖−𝛽𝑗)
 (𝛼𝑖)

𝛼𝑖𝑛 ∏ (

𝑎𝑖
𝑘

+𝑡𝛼𝑖+𝑙−1

𝛼𝑖
)𝑛

𝛼𝑖
𝑙=1

𝑟
𝑖=1  

∏ (𝛽𝑗)
𝛽𝑗𝑛𝑠

𝑗=1  ∏ (

𝑏𝑗
𝑘

+𝑚−1

𝛽𝑗
)𝑛

𝛽𝑗
𝑚=1

 

Let  

𝑎𝑖
𝑘

+𝑡𝛼𝑖+𝑙−1

𝛼𝑖
 = 𝜇𝑙  𝑎𝑛𝑑 

𝑏𝑗

𝑘
+𝑚−1

𝛽𝑗
 = 𝜗𝑚   then,  

𝐷 ≡
∏ 𝑝

𝑛(𝛼𝑖−𝛽𝑗)
 (𝛼𝑖)

𝛼𝑖𝑛𝑟
𝑖=1

∏ (𝛽𝑗)
𝛽𝑗𝑛𝑠

𝑗=1

 ∏ ∏
𝛤(𝜇𝑙+𝑛)

𝛤(𝜇𝑙)

𝛽𝑗

𝑚=1
        

𝛤𝜗𝑚

𝛤𝜗𝑚+𝑛
   

𝛤(𝜗𝑚−𝜇𝑙)

𝛤(𝜗𝑚−𝜇𝑙)

𝛼𝑖
𝑙=1                     (20) 

Put equation (20) in equation (19) and also using beta function we get, 

𝐷 = ∑∞
𝑛=0

∏ 𝑝
𝑛(𝛼𝑖−𝛽𝑗)

𝛤𝑘 (𝑎𝑖) 𝛼𝑖
𝑛𝛼𝑖𝑝

𝑟
𝑖=1

∏ 𝛤𝑘 (𝑏𝑖)𝑝
𝑠
𝑗=1 𝛽𝑗

𝑛𝛽𝑗
∏ ∏

𝛤𝜗𝑚

𝛤𝜗𝑚−𝜇𝑙𝛤𝜇𝑙
∫ 𝑈𝜇𝑙+𝑛−1(1 − 𝑈)𝜗𝑚−𝜇𝑙−1𝑑𝑈 
1

0

𝛽𝑗

𝑚=1

𝛼𝑖
𝑙=1   

above equation can be written in the form of beta function as, 

 𝐷 =
∏ 𝛤𝑘 (𝑎𝑖)𝑝

𝑟
𝑖=1

∏ 𝛤𝑘 (𝑏𝑖)𝑝
𝑠
𝑗=1

   ∏ ∏
1

𝐵(𝜇𝑙,𝜗𝑚−𝜇𝑙)
∫ 𝑈𝜇𝑙−1(1 − 𝑈)𝜗𝑚−𝜇𝑙−1 𝑒

(𝛼𝑖𝑝)𝛼𝑖𝑈𝑧

(𝑝𝛽𝑗)
𝛽𝑗

𝑑𝑈 
1

0

𝛽𝑗

𝑚=1

𝛼𝑖
𝑙=1         

Theorem 6. Let {0}, − +Rkp ; 𝑡 ∈ 𝑵𝟎 Cz , )1,2,...,=;1,2,...,=0;,(, sjriR jiji    



Communications on Applied Nonlinear Analysis 

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and ,\)(),( −++ kZCnbna jjii   then, 

𝛹𝑠
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗, 𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝 = 

 

𝑝
∑

𝑎𝑖
𝑘

𝑟
𝑖=1

𝑘𝑟 ∫ 𝑒−𝑈𝑈
∑ (

𝑎𝑖
𝑘

)−𝑟𝑟
𝑖=1    

∞

0
𝛹𝑠

𝑘 [

−  ;

𝑧(𝑝𝑈)
∑ (

𝛼𝑖𝑡

𝑘
𝑟
𝑖=1 )

(𝑏𝑗 , 𝛽𝑗)1,𝑠;

]0
𝑡,𝑝

dU             (21) 

Proof: Using the definition of t- Generalized p-k Wright function, we have,  

𝛹𝑠
𝑘 [

(𝑎𝑖, 𝛼𝑖𝑡)1,𝑟;

𝑧
(𝑏𝑗 , 𝛽𝑗)1,𝑠;

]𝑟
𝑡,𝑝

 

 = ∑
∏ 𝛤𝑘𝑝 (𝑎𝑖+𝛼𝑖(𝑛+𝑡)   𝑧𝑛𝑟

𝑖=1

∏ 𝛤𝑘𝑝 (𝑏𝑗+𝛽𝑗(𝑛+𝑡)   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

 =  ∑
∏

𝑝

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘

𝑘
  𝛤(

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘
)  𝑧𝑛𝑟

𝑖=1

∏  𝛤𝑘 (𝑏𝑗+𝛽𝑗𝑛)𝑝   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  

                    =  ∑
∏

𝑝

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘

𝑘
     𝑧𝑛𝑟

𝑖=1

∏  𝛤𝑘 (𝑏𝑗+𝛽𝑗𝑛)𝑝   𝑠
𝑗=1 (𝑛+𝑡)!

∞
𝑛=0  ∫ 𝑒−𝑈𝑈

𝑎𝑖+𝛼𝑖(𝑛+𝑡)

𝑘
  −1𝑑𝑈

∞

0
 

 =
𝑝

∑
𝑎𝑖
𝑘

𝑟
𝑖=1

𝑘𝑟  ∫ 𝑒−𝑈𝑈
∑ (

𝑎𝑖
𝑘

)𝑟
𝑖=1  −𝑟    𝛹𝑠

𝑘 [

−  ;

𝑧(𝑝𝑈)
∑ (

𝛼𝑖𝑡

𝑘
𝑟
𝑖=1 )

(𝑏𝑗 , 𝛽𝑗)1,𝑠;

] 𝑑𝑈.0
𝑡,𝑝∞

0
 

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