DOI : 10.17577/IJERTV14IS070052
- Open Access
- Authors : Dr. Amit Mathur, Keshav Charan Pareek
- Paper ID : IJERTV14IS070052
- Volume & Issue : Volume 14, Issue 07 (July 2025)
- Published (First Online): 16-07-2025
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Evaluation of Unified Integrals Involving Products of Generalized M-Series and Incomplete H-Functions
Dr. Amit Mathur
Department of Mathematics, Maulana Azad University, Jodhpur, India
Keshav Charan Pareek
Department of Mathematics, Maulana Azad University, Jodhpur, India
Abstract:
a , x;a , a ;
1
This work presents a unified approach to evaluate a class of j p 2, p z
definite integrals involving the product of an M-series and an
p q b , b ;
incomplete H-function. These integrals are evaluated in terms of
1
q 1,q
the incomplete H-function, yielding generalized and unified q p
expressions. Several special cases are derived by specifying the parameters of the M-series and the incomplete H-function, which
bj a1 l aj l l
j 1 j 2 z
include the Fox H-function, incomplete Fox-Wright functions and incomplete generalized hypergeometric functions. The unified results presented here are broad in scope and hold
p
l
aj
j 1
l 0
q
bj l
j 1
significant applicability in various fields such as science, q p
engineering, and finance.
bj
1 j 1
a1 , xa1
j2 z d ;
Keywords: M-series, Incomplete H-function, Improper integral.
2 i p q
-
INTRODUCTION AND PRELIMINARIES
This section provides a brief overview of essential definitions and notations have been investigated in several prior studies
1 8 related to incomplete function, M-series, and Unified
aj
j 1
arg
and
bj
j 1
q p
(4)
integral, which are used throughout this work.
a1 , x;aj , ap
; bj
a1 l aj l l
2, p
z j 1 j 2 z
q
p q b , b ; p
l 0 l
Incomplete Gamma Function (IGF)
The standard incomplete gamma function
r, x
and
1
q 1,q
aj
j 1
bj l
j 1
r, x expressed by q p
r, x
x
t r 1et dt ;
r 0; x 0
bj
1 j 1
a1 , xa1
j2 z d ;
0 2 i p q
x
r, x t r1et dt ;
(1)
x 0;r 0 when x 0
(2)
aj
j 1
arg
bj
j 1
(5)
such that their sum yields the complete gamma function:
here is a Mellin-Barnes type contour extending from
r, x r, x r ;
r 0
(3)
i to iwith ,and indented, when necessary to separate the sets of poles of the integral in each case.
Incomplete Generalized Hypergeometric function (IGHF)
The incomplete generalized hypergeometric function
Incomplete H-function
Srivastava et al. 10 (equation (2.1) -(2.4)) define incomplete
p q
and p q introduced by Srivastava et al. 9
through
H-function as follows:
Mellin-Barnes integral representation involving the IGF
r, x and r, x as given below
a1 , A1 , x;aj , Aj
and
(1 a , A , x);(1 a , A )
m,n z m,n z
2, p
1, p
z
1 1 1
j 2, p
p,q p,q
bj , Bj
1,q
p,q 1
0,1;1 bj , Bj
1,q
(9)
a1 , A1 , x;a2 , A2 ,…, ap , Ap 1
a1, A1, x;a1, A1 ;
2, p
m,n z
, x z d,
z
p,q
b , B , b , B ,…, b , B
2 i
-
q
b , B ;
1 1 2 2
-
q
j j
1,q
(ii) Additionally putting x 0
in equation (8), incomplete
where
(6)
Fox-Wright function converges to Fox-Wright function (see for details, [14] ([P. 39, Equation (2.6.11)]):
m n a1 , A1 , 0;aj , Aj ; a1 , A1 ;aj , Aj ;
1 a A, x
b
B 1 a A
2, p
z
2, p z
q p
, x
1 1
j 1
j j j j
j 2
-
q
bj , Bj
1,q ;
p q
bj , Bj
1,q ;
1 bj Bj aj Aj
(10)
j m1
j n1
(iii)
If we take
Aj Bj 1 j 1,…, p, j 1,…q in equation
and
(8) & (9), then the incomplete H-function converges to
a , A , x;a A
IGHF p q
and p q
(see details in Srivastava et al.
z z
m,n m,n
p,q p,q
1 1 j ,
bj , Bj
j 2, p
[9]):a ,1, x;a ,1 ;
1,q
1
j 2, p
z
a1, x; a2 ,…, ap ;
z
p q
b ,1 ;
p q
b ,…, b ;
a1 , A1 , x;a2 , A2 ,…, ap , Ap 1
j
1,q
1 p
m,n z
p,q
b , B , b , B ,…, b , B
2 i
, x z d,
(11)
where
1 1 2 2
-
q
(7)
and
2, p
a1 ,1, x;aj ,1 ;
a , x; a ,…, a ;
j
z 1 2
1 p
p z
m n
1 a1 A1, xbj Bj 1 aj Aj
p q
b ,1 ;
1,q
p q
b ,…, b ;
(12)
, x
j 1
q p
j 2
M-series
1 bj Bj aj Aj
Sharma et al. [15] proposed and examined the generalized M-
j m1
j n1
series in the following form:
The incomplete H-function as defined in equation (6) and (7)
respectively, exist for all x 0 under the same set of
,
e ,…, e ;
,
e1 …er zk
k k ;
conditions and contour specifications as presented in the work
M s 1
r z
M s z f1
… fr
k
r f ,…, f ; r k k
of Kilabs et al. 11 , Mathai and Saxena 12 , and Mathai et al. 13 .
1 r
k 0 , , z
, 0
(13)
The previously mentioned functions admit numerous special cases, a few of which are enumerated below:
(i) By putting m=1, n=p and replacing q by q+1 with
From the table of integration, series, and products by I. S. Gradshteyn, M. I. Ryzhik the following Integral formula ([17],
p. 377 Equation (3.257)) is given as:
relevant parameters, the functions (6) & (7) converges
2 1
1
v 2
1
to incomplete Fox-Wright functions
p q
and
p q
uy
w
dy ;
(see for details, [10] [P. 132, Equation (6.3) and (6.4)]):
0
p>y
2uw 2 1
(14)
u 0, v 0, w 0, 1
(1 a1 , A1 , x);(1 a1 , Aj )2, p 2
1, p
z
p,q 1
0,1; 1 bj , Bj
1,q
(8)
-
-
Main Results
a1, A1, x;a1, A1
2, p ;
z
This section presents certain integrals involving the product of
p q
bj , Bj
1,q ;
M-series and Incomplete H-function.
THEOREM 1: Suppose that ; with
R 0
and
By using the result of (14), we get
1 1 1
e1
…er k
k k
z
1
y 0,
u 0,
v 0, w 0, ,
2
u v w
2 2
then
k k
k 0 f1 … fs k
the following integration hold:
k 1
1
1 2
2
,
, x z d
v
e1,…, er ;
z 1 2
uy w M
1
2 i
k 2
y
r s f ,…, f ; v 2
2uw k 1
z
0 1 r uv y w
a , A , x; a , A ,…, a , A
e1
…er k
k k
2 k 0 1 k
1
m,n z2
1 1 2 2
p p dy
k 1
f
… f
k
p,q uv v
w b , B , b , B ,…, b , B
2uw
2
y
s
k
1 1 2 2
q q
k
k 1
k k
1
e1 …er
1 z1
1 2 , x z d
f
… f
k w
2 i k 1 2
2uw
2 k 0 1 k s k
k
z
a , A , x;a , A ,…, a , A ; k 1 ,1
e 1
m, n1
z2 1 1 2 2
p p
2
1 …er
w
p 1,q 1 w
1
k k
b , B ,…, a , B ; k,1
k 0 f1 … fs k
1 1 q q
(15)
2uw
2 k k
k 1
m,n 1
2 z
assume that the conditions for the incomplete H-function p ,q
, x 2 d
in equation [6] are fulfilled.
2 i
k 1
w
Proof: L.H.S. of equation (15) Let
2
1
Through the interpretation of equation (6), the desired result is
derived
v
k k
L uy w
, e1 ,…, er ;
z1 k
r M s 2
e1 …er
1 z1
y
k k
0
y
f1 ,…, fr ; uv v
w
2uw
1
2 k 0 1
… fs
k w
z a1 , A1 , x;a2 , A2 ,…, ap , Ap
1
f
m,n 2
dy
a , A , x;a , A ,…, a , A ;
k ,1
p,q
v 2
b , B , b , B ,…, b , B
m, n1
z2
1 1 2 2
-
p
2
uv y w
1 1 2 2
q q
p 1,q 1 w
By the using of equation (6) and (13) in L we get
b1 , B1 ,…, aq , Bq ; k,1
1 v 2
2 i y
L uy
0
1
w
Hence Theorem 1 is proved.
THEOREM 2: Suppose that ; with R 0
and
k k
e1 …er
1 v 2
z uy
w
1 k
y 0,
u 0,
v 0, w 0, 1 ,
2
1 u v w 1
2 2
then
f
… f
k 1
y
the following integration hold:
k 0 1 k s k
2
1
2
1
v
uy w
, e1 ,…, er ;
z1
v
r M s 2
z2 uy y
w
, x ddy
0
y f1 ,…, fr ; uv v
w
k
y
z a1 , A1 , x;a2 , A2 ,…, ap , Ap
e1 …er z
m,n 2
2
k k 1
p,q
-
v
b , B , b , B ,…, b , B
dy
k k
k 0 f1
… fs
k
uv y w
1 1 2 2
-
-
q
1
v 2
1k
e
…e
1 z k
uy
w
, x z ddy
1 k r k 1
2 i y 2
1 k 0 f
… fs
k w
1
0 2uw 2 k k
Now by changing the order of integration, we get
a , A , x;a , A ,…, a , A ; k 1 ,1
e
…e zk
m, n1
z2
1 1 2 2
-
p
2
1 k r k 1
p 1,q 1 w
k k
2
k 0 f1 … fs k
b1 , B1 ,…, aq , Bq ; k,1
1
2 i
, x z
uy
0
v 2
y
1k
w
dyd
(16)
p , q
assume that the conditions for the incomplete H-function m,n
in equation [7] are fulfilled.
Proof: L.H.S. of equation (16) Let
e
…e
z k
1
2
1
1
1 k r k w
v
,
e ,…, e ; z
k 0 f1 … fs k
L uy
w
M 1
r 1
2uw
y
r s f ,…, f ;
v 2
2 k k
0 1
r uv y
w
k 1
1 2 z
a , A , x;a , A ,…, a , A
, x 2 d
m,n z2
1 1 2 2
p p dy
2 i
k 1
w
p,q uv v
w b , B , b , B ,…, b , B
2
y
1 1 2 2
-
q
Through the interpretation of equation (7), the desired result is
By the using of equation (7) and (13) in L we get
derived
2 1
e
…e
1 z k
1 v
1 k r k 1
L uy
w
1 k 0 f
… f
k w
2 i 0 y
2uw 2
1 k s k
2 1 k
a , A , x;a , A ,…, a , A ; k 1 ,1
e1 …er 1
v
m, n1
z2
1 1 2 2
p p
2
k k
z1 uy
-
w
p 1,q 1 w
k 0 f1 … fs k
y
b , B ,…, a , B ; k,1
k k
-
1 q q
2
1
Hence the Theorem 2 is proved.
z uy v w
, x ddy
2 y
e1 …er z
k
k k 1
k k
k 0 f1 … fs k
-
Special Cases
We illustrate in this part some significant special cases
corresponding to the principal result of Theorem 1 and Theorem 2.
1
uy
v 2
1k
-
-
-
w
, x z ddy
0
2 i y 2
Now by changing the order of integration, we get
e1 …er zk
Corollary 1.
For with 0
and y>0, u>0, v<0, w>0, >- 1
2
k k 1
k k
k 0 f1 … fs k
Furthermore, setting m 1, p n and replacing q with q 1,
in equation (6) and (7), and assume that the incomplete H-
1
2 i
, x z
uy
0
v 2
y
1k
-
w
dyd
function reduces to the incomplete Wright function as given in equation (8) and (9), we obtain the result here from those in Theorems 1 and 2. The corresponding integral formula as
2
By using the result of (14), we get given:
e1
…er zk
1
2
k k 1
v
,
e1,…, er ; z
f … f k
uy
-
w M
1
k 0 1 k s k
y
r s f ,…, f ;
v 2
1
0 1
r uv y
-
w
2
1 k 2
, x z d
z a1 , A1 , x;a2 , A2 ,…, ap , Ap
2
2 i
k 1
-
p q
dy
2uw k 1
uv v
w b , B , b , B ,…, b , B
y
2
1 1 2 2
q q
e1 …er zk k
k k
k k 1
1
e1 …er
w
1 z1
2 k k
k k 0 f1 … fs k
1
k 0 f1
… fs
k
2uw
k 1
2uw 2 k k
1
1 2
z
a1 , A1 , x;a2 , A2 ,…, ap , Ap ; k ,1
, x z d
2
2
2 i k 1 2
and
p 1
w
q 1
b1 , B1 ,…, aq , Bq ; k 1,1
(17)
M
2
1
e
…e
1 z k
v
,
e1,…, er ; z
1 k r k 1
uy
w 1
f
… f
k w
r s
2
y
f1 ,…, fr ;
v
2uw
2 k 0 1 k s k
0
uv y
-
-
w
1
z
a , x; a ,…, a ; k 1
z a1 , A1 , x;a2 , A2 ,…, ap , Ap
2 1 2 p
2
2
dy
p 1
q 1
p q uv v
w b , B , b , B ,…, b , B
w b ,…, b ; k
2
y
e
…e
1 1 2 2 q q
1 z k
1 p
(20)
1 k r k 1
under the assumption that each element of equation (19) and
1 k 0 f
… f
k w
2uw 2
1 k s k
(20) are exists.
a , A , x;a , A ,…, a , A ; k 1 ,1 (18)
Corollary 3:
z2
1 1 2 2
p p
2
p 1 q 1 w
1
b1 , B1 ,…, aq , Bq ; k 1,1
For with 0 and y>0, u>0, v<0, w>0, >-
under the assumption that each element of equation (17) and
2
Furthermore, setting x 0 , in equation (8), and assume that
(18) are exists.
Corollary 2:
For with 0
and y>0, u>0, v<0, w>0, 1
>- .
2
the incomplete H-function reduces to the incomplete Fox- Wright generalized hypergeometric function as given in equation (10), also M-series reduces into unity, we obtain the result here from those in Theorems 1. The corresponding integral formula as given:
Furthermore, setting Aj Bj 1, j 1,…, p; j 1,…, q
v 2
1
in equation (8) and (9), and assume that the incomplete H- function reduces to the incomplete generalized
uy
y
0
w
hypergeometric function as given in equation (11) and (12),
a , A ,…, a , A ;
v 2
1
1 1
p p
z uy w
dy
we obtain the result here from those in Theorems 1 and 2. The
p q b , B ;
y
corresponding integral formula as given:
j j
1,q
v 2
1
w
,
M
e ,…, e ; z
1
uy
y
r
r s f ,…, f ;
v 2
1 p 1 q 1
2
1
1
z a1 , A1 ,…, ap , Ap ;
,1
0 1
r uv y
w
w
2uw 2
b1 , B1 ,…, aq , Bq ; 1,1
a , x; a ,…, a ;
1 2
1 p
p z2
uv y
dy
w
(21)
p q
b ,…, b ;
v 2
e
…e
1 z k
under the assumption that each element of equation (21) are
exists.
1 k r k 1
1 k 0 f
… f
k w
-
CONCLUSION
2uw 2 1 k s k
z a , x; a ,…, a ; k 1
In the present study, we have established several significant integrals involving the product of the M-series and incomplete
2 1 2 p
2
H-functions, represented in terms of incomplete H-functions
p 1
q 1 w
themselves. Additionally, we have presented certain special
and
2 1
b1,…, bp ; k
(19)
cases by assigning certain values to the parameters of the M- series and incomplete H-functions-such as Foxs H-function, incomplete FoxWright functions, FoxWright functions, and
incomplete generalized hypergeometric functions. Some
v
uy w
, e1 ,…, er ;
z1
previously known results are also included as special cases.
y r M s f ,…, f ;
v 2
The integrals derived in this analysis are of a general form and
0
1
r uv y w
may serve as a foundation for developing numerous results
a , x; a ,…, a ; z
relevant to practical and applied contexts
1 2
p 2 dy
1 p
uv y w
p q
b ,…, b ;
v 2
REFERENCES
-
M.K. Bansal, D. Kumar, I. Khan, J. Singh and K. S. Nisar, Certain Unified Integrals Associated with Product of M-Series and Incomplete H- functions, Mathematics, 7(12), 1191, (2019).
-
Bansal, M.K.; Choi, J. A Note on Path way Fractional Integral Formulas Associated with the Incomplete
H-functions. Int. J. Appl. Comput. Math. 2019, 5, 133.
-
Bansal, M.K.; Jolly, N.; Jain, R.; Kumar, D. An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results. J. Anal. 2019, 27, 727740.
-
Chaurasia, V.B.L.; Kumar, D. The integration of certain product involving special functions. Sci. Ser. A Math. Sci. 2010, 19, 712.
-
Chaurasia, V.B.L.; Singh, J. Certain integral properties pertaining to special functions. Sci. Ser. A Math. Sci. 2010,
19, 16.
-
Kumar, D.; Ayant, F.; Kumar, D. A new class of integrals involving generalized hypergeometric function and multivariable Aleph-function. Kragujevac J. Math. 2020, 44, 539550.
-
Brychkov, Y.A. Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas; CRC Press:
Boca Raton, FL, USA, 2008.
-
Garg, M.; Mittal, S. On a new unified integral. Proc. Indian Acad. Sci. Math. Sci. 2004, 114, 99101.
-
Srivastava, H.M.; Chaudhary, M.A.; Agarwal, R.P. The Incomplete Poch hammer Symbols and Their Applications to Hypergeometric and Related Functions. Integral Transform. Spec. Funct., 2012, 23, 659683.
-
Srivastava, H.M.; Saxena, R.K.; Parmar, R.K. Some Families of
the Incomplete H-functions and the Incomplete
Hfunctions and Associated Integral Transforms and Operators of Fractional Calculus with Applications. Russ. J. Math. Phys. 2018, 25, 116138.
-
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theorem and Applications of Fractional Differential Equations; (North-Holland Mathematical Studies); Elsevier (North Holland) Science Publishers: Amsterdam, The Netherland; London, UK; New York, NX, USA, 2006; p. 204.
-
Mathai, A.M.; Saxena, R.K. The H-Function with Applications in Statistics Other Disciplines; Wiley Eastern: New Delhi, India; Wiley Halsted: New York, NX, USA, 1978.
-
Mathai, A.M.; Saxena, R.K.; Haubold, H.J. The H-function: Theorem and Applications; Springer: New York, NX, USA, 2009.
-
Srivastava, H.M.; Gupta, K.C.; Goyal, S.P. The H-Functions of One and Two Variables with Applications; South Asian Publishers: New Delhi, India, 1982.
-
Sharma, M.; Jain, R. A Note on a Generalized M-Series as a Special Function of Fractional Calculus. Frac., Calc. Appl. Anal. 2009, 12, 449452.
-
Choi, J.; Hasanov, A.; Srivastava, H.M.; Turaev, M. Integral representations for Srivastavas triple hypergeometric functions. Taiwan. J. Math. 2011, 15, 27512762.
-
Gradshteyn, I.S. and Ryzhik, I.M., Table of Integrals, Series and product, Fourth Editions 1965, Enlarged Edition by A. Jeffrey Academic press, New York 1994.
