SlideShare a Scribd company logo
1 of 4
Download to read offline
IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 16-19
www.iosrjournals.org
www.iosrjournals.org 16 | Page
I- Function and H -Function Associated With Double Integral
Ashok Singh Shekhawat*
, Rakeshwar Purohit**
And Jyoti Shaktawat***
*Department of Mathematics, Regional college for Education, Research and Technology,
Jaipur, Rajasthan (India)
**Department of Mathematics and Statistics, University College of Science, Mohan Lal Sukhadia University,
Udaipur, Rajasthan (India)
***
Department of Mathematics, Kautilya Institute of Technology and Engineering,
Jaipur, Rajasthan (India)
Abstract: The object of this paper is to discuss certain integral properties of a I -function and H -function,
proposed by Inayat-Hussain which contain a certain class of Feynman integrals, the exact partition of a Gaussian
model in Statistical Mechanics and several other functions as its particular cases. During the course of finding,
we establish certain new double integral relation pertaining to a product involving I function and H function.
These double integral relations are unified in nature and act as a key formulae from which we can obtain as their
special case, double integral relations concerning a large number of simple special functions. For the sake of
illustration, we record here some special cases of our main results which are also new and of interest by
themselves. All the result which are established in this paper are basic in nature and are likely to find useful
applications in several fields notably electrical network, probability theory and statistical mechanics.
Key words. I function, H -function, Hermite polynomials, Laguerre polynomials.
I. Introduction
The H -function [6] is a new generalization of the well known Fox’s H-function [4]. The H -function
pertains the exact partition function of the Gaussian model in statistical mechanics, functions useful in testing
hypothesis and several others as its particular cases. The conventional formulation may fail pertaining to the
domain of quantum cosmology but Feynman path integrals apply [10,11]. Feynman integral are useful in the
study and development of simple and multiple variable hypergeometric series which in turn are useful in
statistical mechanics.
The I-function defined as









jijiajja
jijibjj(b
zII[z]











zI ip1,njijian1,jja
iq1,mjijibm1,jj{(b
nm,
iqip 
dszs)
2
1 s


 
where















s)as)b1
s)a1s)b
s)
jiji
ip
1nj
jiji
iq
1mj1i
jj
n
1j
jj
m
1j

…(1.1)
The H -function will be defined and represent as given in [1]








 



  dx
i2
1
xHx]H
i
i
P1,NjjaN1,jAjj(a
Q1,MjjbM1,jj(b
NM,
QP,
NM,
QP,
…(1.2)
where
I- Function And H -Function Associated With Double Integral
www.iosrjournals.org 17 | Page







jj
P
1Nj
jj
Q
1Mj
jjj
N
1j
jj
M
1j
ab1
Aa1b
…(1.3)
which contains fractional powers of some of the gamma functions. Here aj (j = 1,…,P) and bj (j =
1,…,Q) are complex parameters, aj  0 (j = 1,…,P), j  0 (j = 1,…,Q) (not all zero simultaneously and the
exponents Aj (j = 1,…,N) and Bj (j = M+1,…,Q) can take on non-integer values. The contour in (1.2) is
imaginary axis R() = 0. It is suitably indented in order to avoid the singularities of the gamma functions and to
keep those singularities on appropriate side. Again for Aj (j=1,…,N) not an integer, the poles of the
gamma function of the numerator in (1.3) are converted to branch points. However, a long as there is no
coincidence of pole from any N)1,...,ja1andM)1,...,jb jjjj
 pair, the
branch cuts can be chosen so that the path of integration can be distorted in the useful manner. For the sake of
brevity
0BAT j
P
1Nj
jj
Q
1Mj
jj
N
1j
j
M
1j
 

II. Main Result
We will obtain the following result:
(A) dydx
xy1
yv1
Hvy
xy1
x1
I
y)x)(1(1
xy1
xy1
y1
vy
xy1
x1 NM,
QP,
1
0
1
0



























































vHvIs
P1,NjjaN1,jAjja11
1k1Q1,MjBjjbM1,jjb
1NM,
1Q1,P
jijiajja
jijibjj(b
…(2.1)
provided that  T
2
1
|Varg0b[R jj
Proof. We have
































s)as)b1
sa1s)b
2
1
v
xy1
y)1
Hvy
xy1
x1
I
jiji
ip
1nj
jiji
iq
1mj1i
jj
n
1j
jj
m
1jNM,
QP, 
. 
























 dv
xy1
y1
ab1
Aa1b
i2
1
dsvy
xy1
x1
jj
P
1Nj
jjj
Q
1Mj
jjj
N
1j
jj
M
1j
i
i
s
…(2.2)
Multiplying both sides of (2.2) by 
























y)x)(1(1
xy1
xy1
y1
y
xy1
x1
and integration with respect to x and y between 0 and 1 for both the variable and making a use of a known result
[2, p.145],we get the required result (2.1) after a little simplification
I- Function And H -Function Associated With Double Integral
www.iosrjournals.org 18 | Page
(B) dvduv]HI(u)uvv)u
NM,
QP,
11
00
 


dzzz)zIs 1
0
jijiajja
jijibjj(b









 
. dzzH
P1,NjjaN1,jAjja11
1s1Q1,MjBjjbM1,jjb
1NM,
1Q1,P







 



…(2.3)
provided that  0bR( jj
Proof. Using (1.1) and (1.2), we have
dsu
s)as)b1
sa1s)b
2
1
v]HI(u) s
jiji
ip
1nj
jij
iq
1mj1i
jj
n
1j
jj
m
1jNM,
QP,


















. 










 dv
ab1
Aa1b
i2
1
jj
P
1Nj
jjj
Q
1Mj
jjj
N
1j
jj
M
1j
i
i
…(2.4)
Multiplying both side by
11
uvv)u 
 and integrating with respect to u and v between 0 and
 for both the variable and make a use of a known result [2, p.177], we get the required result. Letting
pz
ez) 
 in (2.3), we get the particular case after simplification
(c) dvduv]1Hu)]I[(v(1vv)1u)(1f(uv)
NM,
QP,
11
1
0
1
0
 

1s
1
0
jijiajja
jijibjj(b
z)(1f(z)z)(1Is 







 
. dzz)1H
P1,NjjaN1,jAjja11
1s1Q1,MjBjjbM1,jjb
1NM,
1Q1,P













…(2.5)
provided that R() > 0, R() > 0.
Proof. Using equation (1.1) and (1.2), we have
dsu)1v
s)as)b1
sa1s)b
2
1
v]1Hu)]I[v(1 ss
jiji
ip
1nj
jij
iq
1mj1i
jj
n
1j
jj
m
1jNM,
QP,



















I- Function And H -Function Associated With Double Integral
www.iosrjournals.org 19 | Page
. 










 dv)1
ab1
Aa1b
i2
1
jj
P
1Nj
jjj
Q
1Mj
jjj
N
1j
jj
M
1j
i
i
…(2.6)
Multiplying both side of (2.6) by

 vv)1u)(1f(uv) 11
and integrating with respect to u and v
between 0 and 1 for both the variable and use of result [2, p.243] and by further simplification, we get the result
(2.5).
Letting f(z) =
1
z 
in (2.5), we get the particular result after simplification.
Particular Case
(i) Taking 















vy
xy1
x1
Svy
xy1
x1
I m
n
The result in (2.1), (2.2) and (2.3) reduces to the known result after a slight simplification obtained by Chaurasia
and Shekhawat [2].
Acknowledgement.
The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and
valuable suggestions in the preparation of this paper.
References
[1]. R.G. Buschman and H.M. Srivastava, The H -function associated with a certain class of Feynman integrals, J. Phys. A: Math. Gen..
23 (1990), 4707-4710.
[2]. V.B.L. Chaurasia and Ashok Singh Shekhawat, Some integral properties of a general class of polynomials associated with Feynman
integrals, Bull. Malaysian Math. Sc. Soc. (2) (28) (2) (2005), 183-189.
[3]. J. Edewards, A Treatise on integral calculus, Chelsea Pub. Co., 2 (1922).
[4]. C. Fox, The G and H-functions as Symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429.
[5]. C. Grosche and F. Steiner, Hand Book of Feynman Path integrals, Springer Tracts in Modern Physics Vol.145, Springer-Verlag Berlin
Heidelberg, New York, 1998.
[6]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals : I. Transformation and reduction
formulae, J. Phys. A: Math. Gen. 20 (1987), 4109-4117.
[7]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals :II. A generalization of the H-
function, J. Phys. A: Math. Gen. 20 (1987), 4119-4128.
[8]. Saxena, R.K., On fractional integer operators, Math. Zeitschr, 96 (1967), 288-291.
[9]. Saxena, R.K. and Gupta, N., Some Abelian theorems for distributional H-function Transformation, Indian J. Pure Appl. Math. 25(8)
(1994), 869-879.
[10]. H.M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14 (1972), 1-6.
[11]. H.M. Srivastava and N.P. Singh, The integration of certain products of Multivariable H-function with a general class of Polynomials,
Rend. Circ. Mat. Palermo 2(32) (1983), 157-187.
[12]. C. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23 Fourth edition, Amer. Math. Soc. Providence, Rhode Island
(1975).

More Related Content

What's hot

Paper id 21201486
Paper id 21201486Paper id 21201486
Paper id 21201486IJRAT
 
Characteristic orthogonal polynimial application to galerkin indirect variati...
Characteristic orthogonal polynimial application to galerkin indirect variati...Characteristic orthogonal polynimial application to galerkin indirect variati...
Characteristic orthogonal polynimial application to galerkin indirect variati...eSAT Publishing House
 
Auto Regressive Process (1) with Change Point: Bayesian Approch
Auto Regressive Process (1) with Change Point: Bayesian ApprochAuto Regressive Process (1) with Change Point: Bayesian Approch
Auto Regressive Process (1) with Change Point: Bayesian ApprochIJRESJOURNAL
 
A Class of Polynomials Associated with Differential Operator and with a Gener...
A Class of Polynomials Associated with Differential Operator and with a Gener...A Class of Polynomials Associated with Differential Operator and with a Gener...
A Class of Polynomials Associated with Differential Operator and with a Gener...iosrjce
 
FEA Quiz - worksheets
FEA Quiz - worksheetsFEA Quiz - worksheets
FEA Quiz - worksheetsgokulfea
 
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...Rene Kotze
 
Bianchi type i wet dark universe in bimetric relativity
Bianchi type i  wet  dark  universe in bimetric relativityBianchi type i  wet  dark  universe in bimetric relativity
Bianchi type i wet dark universe in bimetric relativityAlexander Decker
 
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
 
International journal of engineering and mathematical modelling vol2 no1_2015_1
International journal of engineering and mathematical modelling vol2 no1_2015_1International journal of engineering and mathematical modelling vol2 no1_2015_1
International journal of engineering and mathematical modelling vol2 no1_2015_1IJEMM
 
The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...
The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...
The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...ijtsrd
 
Correlation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi setsCorrelation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
 
Correlation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi setsCorrelation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi setseSAT Publishing House
 
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
 
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...IRJESJOURNAL
 

What's hot (17)

Paper id 21201486
Paper id 21201486Paper id 21201486
Paper id 21201486
 
Characteristic orthogonal polynimial application to galerkin indirect variati...
Characteristic orthogonal polynimial application to galerkin indirect variati...Characteristic orthogonal polynimial application to galerkin indirect variati...
Characteristic orthogonal polynimial application to galerkin indirect variati...
 
Auto Regressive Process (1) with Change Point: Bayesian Approch
Auto Regressive Process (1) with Change Point: Bayesian ApprochAuto Regressive Process (1) with Change Point: Bayesian Approch
Auto Regressive Process (1) with Change Point: Bayesian Approch
 
Bc4301300308
Bc4301300308Bc4301300308
Bc4301300308
 
ED7201 FEMMD_notes
ED7201 FEMMD_notesED7201 FEMMD_notes
ED7201 FEMMD_notes
 
Project 7
Project 7Project 7
Project 7
 
A Class of Polynomials Associated with Differential Operator and with a Gener...
A Class of Polynomials Associated with Differential Operator and with a Gener...A Class of Polynomials Associated with Differential Operator and with a Gener...
A Class of Polynomials Associated with Differential Operator and with a Gener...
 
FEA Quiz - worksheets
FEA Quiz - worksheetsFEA Quiz - worksheets
FEA Quiz - worksheets
 
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch) TITLE: Path...
 
Bianchi type i wet dark universe in bimetric relativity
Bianchi type i  wet  dark  universe in bimetric relativityBianchi type i  wet  dark  universe in bimetric relativity
Bianchi type i wet dark universe in bimetric relativity
 
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"
 
International journal of engineering and mathematical modelling vol2 no1_2015_1
International journal of engineering and mathematical modelling vol2 no1_2015_1International journal of engineering and mathematical modelling vol2 no1_2015_1
International journal of engineering and mathematical modelling vol2 no1_2015_1
 
The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...
The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...
The Super Stability of Third Order Linear Ordinary Differential Homogeneous E...
 
Correlation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi setsCorrelation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi sets
 
Correlation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi setsCorrelation measure for intuitionistic fuzzy multi sets
Correlation measure for intuitionistic fuzzy multi sets
 
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...
 
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...
Buckling of a carbon nanotube embedded in elastic medium via nonlocal elastic...
 

Viewers also liked

J010627176.iosr jece
J010627176.iosr jeceJ010627176.iosr jece
J010627176.iosr jeceIOSR Journals
 
Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...
Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...
Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...IOSR Journals
 
m - projective curvature tensor on a Lorentzian para – Sasakian manifolds
m - projective curvature tensor on a Lorentzian para – Sasakian manifoldsm - projective curvature tensor on a Lorentzian para – Sasakian manifolds
m - projective curvature tensor on a Lorentzian para – Sasakian manifoldsIOSR Journals
 
Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...
Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...
Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...IOSR Journals
 
A Study On Psychological Variables On Women Sports Participation Levels In Un...
A Study On Psychological Variables On Women Sports Participation Levels In Un...A Study On Psychological Variables On Women Sports Participation Levels In Un...
A Study On Psychological Variables On Women Sports Participation Levels In Un...IOSR Journals
 
Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...
Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...
Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...IOSR Journals
 
Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...
Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...
Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...IOSR Journals
 
Energy Policy in India
Energy Policy in IndiaEnergy Policy in India
Energy Policy in IndiaIOSR Journals
 
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive Radio
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive RadioFractal Reconfigurable Multiband Communicating Antenna for Cognitive Radio
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive RadioIOSR Journals
 

Viewers also liked (20)

J010627176.iosr jece
J010627176.iosr jeceJ010627176.iosr jece
J010627176.iosr jece
 
D010121832
D010121832D010121832
D010121832
 
Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...
Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...
Molecular characterization of pea (Pisum sativum L.) using microsatellite mar...
 
m - projective curvature tensor on a Lorentzian para – Sasakian manifolds
m - projective curvature tensor on a Lorentzian para – Sasakian manifoldsm - projective curvature tensor on a Lorentzian para – Sasakian manifolds
m - projective curvature tensor on a Lorentzian para – Sasakian manifolds
 
Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...
Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...
Corporate Governance, Firm Size, and Earning Management: Evidence in Indonesi...
 
C1803061620
C1803061620C1803061620
C1803061620
 
A Study On Psychological Variables On Women Sports Participation Levels In Un...
A Study On Psychological Variables On Women Sports Participation Levels In Un...A Study On Psychological Variables On Women Sports Participation Levels In Un...
A Study On Psychological Variables On Women Sports Participation Levels In Un...
 
Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...
Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...
Levels of PAHs and Potentially Toxic Metals in Three Species of Fresh and Smo...
 
Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...
Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...
Marketing Analysis of onion in Bade and Geidam Local Government Areas of Yobe...
 
Energy Policy in India
Energy Policy in IndiaEnergy Policy in India
Energy Policy in India
 
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive Radio
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive RadioFractal Reconfigurable Multiband Communicating Antenna for Cognitive Radio
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive Radio
 
C010311421
C010311421C010311421
C010311421
 
H010315456
H010315456H010315456
H010315456
 
E011113336
E011113336E011113336
E011113336
 
G011123539
G011123539G011123539
G011123539
 
I1304026367
I1304026367I1304026367
I1304026367
 
P01226101106
P01226101106P01226101106
P01226101106
 
L010127578
L010127578L010127578
L010127578
 
J017367075
J017367075J017367075
J017367075
 
N010237982
N010237982N010237982
N010237982
 

Similar to I- Function and H -Function Associated With Double Integral

On Some New Linear Generating Relations Involving I-Function of Two Variables
On Some New Linear Generating Relations Involving I-Function of Two VariablesOn Some New Linear Generating Relations Involving I-Function of Two Variables
On Some New Linear Generating Relations Involving I-Function of Two VariablesIOSRJM
 
On a Deterministic Property of the Category of k-almost Primes: A Determinist...
On a Deterministic Property of the Category of k-almost Primes: A Determinist...On a Deterministic Property of the Category of k-almost Primes: A Determinist...
On a Deterministic Property of the Category of k-almost Primes: A Determinist...Ramin (A.) Zahedi
 
Vibration analysis and response characteristics of a half car model subjected...
Vibration analysis and response characteristics of a half car model subjected...Vibration analysis and response characteristics of a half car model subjected...
Vibration analysis and response characteristics of a half car model subjected...editorijrei
 
International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
 
The Generalized Difference Operator of the 퐧 퐭퐡 Kind
The Generalized Difference Operator of the 퐧 퐭퐡 KindThe Generalized Difference Operator of the 퐧 퐭퐡 Kind
The Generalized Difference Operator of the 퐧 퐭퐡 KindDr. Amarjeet Singh
 
Arts revealed in calculus and its extension
Arts revealed in calculus and its extensionArts revealed in calculus and its extension
Arts revealed in calculus and its extensionPremier Publishers
 
An alternative approach to estimation of population
An alternative approach to estimation of populationAn alternative approach to estimation of population
An alternative approach to estimation of populationAlexander Decker
 
Trialdraftsppformat dimen test1
Trialdraftsppformat dimen   test1Trialdraftsppformat dimen   test1
Trialdraftsppformat dimen test1foxtrot jp R
 
PaperNo6-YousefiHabibi-IJAM
PaperNo6-YousefiHabibi-IJAMPaperNo6-YousefiHabibi-IJAM
PaperNo6-YousefiHabibi-IJAMMezban Habibi
 
H function and a problem related to a string
H function and a problem related to a stringH function and a problem related to a string
H function and a problem related to a stringAlexander Decker
 
Comparative Study of the Effect of Different Collocation Points on Legendre-C...
Comparative Study of the Effect of Different Collocation Points on Legendre-C...Comparative Study of the Effect of Different Collocation Points on Legendre-C...
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
 

Similar to I- Function and H -Function Associated With Double Integral (20)

Ramanujan Integral involving the Product of Generalized Zetafunction and - fu...
Ramanujan Integral involving the Product of Generalized Zetafunction and - fu...Ramanujan Integral involving the Product of Generalized Zetafunction and - fu...
Ramanujan Integral involving the Product of Generalized Zetafunction and - fu...
 
513
513513
513
 
lec25.ppt
lec25.pptlec25.ppt
lec25.ppt
 
ON SUB RIEMANNIAN STRUCTURES AND RELATED MECHANICS
ON SUB RIEMANNIAN STRUCTURES AND RELATED MECHANICSON SUB RIEMANNIAN STRUCTURES AND RELATED MECHANICS
ON SUB RIEMANNIAN STRUCTURES AND RELATED MECHANICS
 
On Some New Linear Generating Relations Involving I-Function of Two Variables
On Some New Linear Generating Relations Involving I-Function of Two VariablesOn Some New Linear Generating Relations Involving I-Function of Two Variables
On Some New Linear Generating Relations Involving I-Function of Two Variables
 
D0561416
D0561416D0561416
D0561416
 
Paper 1 (rajesh singh)
Paper 1 (rajesh singh)Paper 1 (rajesh singh)
Paper 1 (rajesh singh)
 
On a Deterministic Property of the Category of k-almost Primes: A Determinist...
On a Deterministic Property of the Category of k-almost Primes: A Determinist...On a Deterministic Property of the Category of k-almost Primes: A Determinist...
On a Deterministic Property of the Category of k-almost Primes: A Determinist...
 
Vibration analysis and response characteristics of a half car model subjected...
Vibration analysis and response characteristics of a half car model subjected...Vibration analysis and response characteristics of a half car model subjected...
Vibration analysis and response characteristics of a half car model subjected...
 
International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI)
International Journal of Mathematics and Statistics Invention (IJMSI)
 
A05920109
A05920109A05920109
A05920109
 
The Generalized Difference Operator of the 퐧 퐭퐡 Kind
The Generalized Difference Operator of the 퐧 퐭퐡 KindThe Generalized Difference Operator of the 퐧 퐭퐡 Kind
The Generalized Difference Operator of the 퐧 퐭퐡 Kind
 
g-lecture.pptx
g-lecture.pptxg-lecture.pptx
g-lecture.pptx
 
Arts revealed in calculus and its extension
Arts revealed in calculus and its extensionArts revealed in calculus and its extension
Arts revealed in calculus and its extension
 
201977 1-1-1-pb
201977 1-1-1-pb201977 1-1-1-pb
201977 1-1-1-pb
 
An alternative approach to estimation of population
An alternative approach to estimation of populationAn alternative approach to estimation of population
An alternative approach to estimation of population
 
Trialdraftsppformat dimen test1
Trialdraftsppformat dimen   test1Trialdraftsppformat dimen   test1
Trialdraftsppformat dimen test1
 
PaperNo6-YousefiHabibi-IJAM
PaperNo6-YousefiHabibi-IJAMPaperNo6-YousefiHabibi-IJAM
PaperNo6-YousefiHabibi-IJAM
 
H function and a problem related to a string
H function and a problem related to a stringH function and a problem related to a string
H function and a problem related to a string
 
Comparative Study of the Effect of Different Collocation Points on Legendre-C...
Comparative Study of the Effect of Different Collocation Points on Legendre-C...Comparative Study of the Effect of Different Collocation Points on Legendre-C...
Comparative Study of the Effect of Different Collocation Points on Legendre-C...
 

More from IOSR Journals (20)

A011140104
A011140104A011140104
A011140104
 
M0111397100
M0111397100M0111397100
M0111397100
 
L011138596
L011138596L011138596
L011138596
 
K011138084
K011138084K011138084
K011138084
 
J011137479
J011137479J011137479
J011137479
 
I011136673
I011136673I011136673
I011136673
 
G011134454
G011134454G011134454
G011134454
 
H011135565
H011135565H011135565
H011135565
 
F011134043
F011134043F011134043
F011134043
 
E011133639
E011133639E011133639
E011133639
 
D011132635
D011132635D011132635
D011132635
 
C011131925
C011131925C011131925
C011131925
 
B011130918
B011130918B011130918
B011130918
 
A011130108
A011130108A011130108
A011130108
 
I011125160
I011125160I011125160
I011125160
 
H011124050
H011124050H011124050
H011124050
 
F011123134
F011123134F011123134
F011123134
 
E011122530
E011122530E011122530
E011122530
 
D011121524
D011121524D011121524
D011121524
 
C011121114
C011121114C011121114
C011121114
 

Recently uploaded

Biogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune Waterworlds
Biogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune WaterworldsBiogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune Waterworlds
Biogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune WaterworldsSérgio Sacani
 
CYTOGENETIC MAP................ ppt.pptx
CYTOGENETIC MAP................ ppt.pptxCYTOGENETIC MAP................ ppt.pptx
CYTOGENETIC MAP................ ppt.pptxCherry
 
Role of AI in seed science Predictive modelling and Beyond.pptx
Role of AI in seed science  Predictive modelling and  Beyond.pptxRole of AI in seed science  Predictive modelling and  Beyond.pptx
Role of AI in seed science Predictive modelling and Beyond.pptxArvind Kumar
 
Cyathodium bryophyte: morphology, anatomy, reproduction etc.
Cyathodium bryophyte: morphology, anatomy, reproduction etc.Cyathodium bryophyte: morphology, anatomy, reproduction etc.
Cyathodium bryophyte: morphology, anatomy, reproduction etc.Cherry
 
Thyroid Physiology_Dr.E. Muralinath_ Associate Professor
Thyroid Physiology_Dr.E. Muralinath_ Associate ProfessorThyroid Physiology_Dr.E. Muralinath_ Associate Professor
Thyroid Physiology_Dr.E. Muralinath_ Associate Professormuralinath2
 
TransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRings
TransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRingsTransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRings
TransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRingsSérgio Sacani
 
Pteris : features, anatomy, morphology and lifecycle
Pteris : features, anatomy, morphology and lifecyclePteris : features, anatomy, morphology and lifecycle
Pteris : features, anatomy, morphology and lifecycleCherry
 
FS P2 COMBO MSTA LAST PUSH past exam papers.
FS P2 COMBO MSTA LAST PUSH past exam papers.FS P2 COMBO MSTA LAST PUSH past exam papers.
FS P2 COMBO MSTA LAST PUSH past exam papers.takadzanijustinmaime
 
The Mariana Trench remarkable geological features on Earth.pptx
The Mariana Trench remarkable geological features on Earth.pptxThe Mariana Trench remarkable geological features on Earth.pptx
The Mariana Trench remarkable geological features on Earth.pptxseri bangash
 
Understanding Partial Differential Equations: Types and Solution Methods
Understanding Partial Differential Equations: Types and Solution MethodsUnderstanding Partial Differential Equations: Types and Solution Methods
Understanding Partial Differential Equations: Types and Solution Methodsimroshankoirala
 
Efficient spin-up of Earth System Models usingsequence acceleration
Efficient spin-up of Earth System Models usingsequence accelerationEfficient spin-up of Earth System Models usingsequence acceleration
Efficient spin-up of Earth System Models usingsequence accelerationSérgio Sacani
 
Digital Dentistry.Digital Dentistryvv.pptx
Digital Dentistry.Digital Dentistryvv.pptxDigital Dentistry.Digital Dentistryvv.pptx
Digital Dentistry.Digital Dentistryvv.pptxMohamedFarag457087
 
Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS ESCORT SERVICE In Bhiwan...
Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS  ESCORT SERVICE In Bhiwan...Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS  ESCORT SERVICE In Bhiwan...
Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS ESCORT SERVICE In Bhiwan...Monika Rani
 
Selaginella: features, morphology ,anatomy and reproduction.
Selaginella: features, morphology ,anatomy and reproduction.Selaginella: features, morphology ,anatomy and reproduction.
Selaginella: features, morphology ,anatomy and reproduction.Cherry
 
module for grade 9 for distance learning
module for grade 9 for distance learningmodule for grade 9 for distance learning
module for grade 9 for distance learninglevieagacer
 
Climate Change Impacts on Terrestrial and Aquatic Ecosystems.pptx
Climate Change Impacts on Terrestrial and Aquatic Ecosystems.pptxClimate Change Impacts on Terrestrial and Aquatic Ecosystems.pptx
Climate Change Impacts on Terrestrial and Aquatic Ecosystems.pptxDiariAli
 

Recently uploaded (20)

Biogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune Waterworlds
Biogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune WaterworldsBiogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune Waterworlds
Biogenic Sulfur Gases as Biosignatures on Temperate Sub-Neptune Waterworlds
 
CYTOGENETIC MAP................ ppt.pptx
CYTOGENETIC MAP................ ppt.pptxCYTOGENETIC MAP................ ppt.pptx
CYTOGENETIC MAP................ ppt.pptx
 
Role of AI in seed science Predictive modelling and Beyond.pptx
Role of AI in seed science  Predictive modelling and  Beyond.pptxRole of AI in seed science  Predictive modelling and  Beyond.pptx
Role of AI in seed science Predictive modelling and Beyond.pptx
 
Cyathodium bryophyte: morphology, anatomy, reproduction etc.
Cyathodium bryophyte: morphology, anatomy, reproduction etc.Cyathodium bryophyte: morphology, anatomy, reproduction etc.
Cyathodium bryophyte: morphology, anatomy, reproduction etc.
 
Thyroid Physiology_Dr.E. Muralinath_ Associate Professor
Thyroid Physiology_Dr.E. Muralinath_ Associate ProfessorThyroid Physiology_Dr.E. Muralinath_ Associate Professor
Thyroid Physiology_Dr.E. Muralinath_ Associate Professor
 
TransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRings
TransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRingsTransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRings
TransientOffsetin14CAftertheCarringtonEventRecordedbyPolarTreeRings
 
PATNA CALL GIRLS 8617370543 LOW PRICE ESCORT SERVICE
PATNA CALL GIRLS 8617370543 LOW PRICE ESCORT SERVICEPATNA CALL GIRLS 8617370543 LOW PRICE ESCORT SERVICE
PATNA CALL GIRLS 8617370543 LOW PRICE ESCORT SERVICE
 
Pteris : features, anatomy, morphology and lifecycle
Pteris : features, anatomy, morphology and lifecyclePteris : features, anatomy, morphology and lifecycle
Pteris : features, anatomy, morphology and lifecycle
 
ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY // USES OF ANTIOBIOTICS TYPES OF ANTIB...
ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY  // USES OF ANTIOBIOTICS TYPES OF ANTIB...ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY  // USES OF ANTIOBIOTICS TYPES OF ANTIB...
ABHISHEK ANTIBIOTICS PPT MICROBIOLOGY // USES OF ANTIOBIOTICS TYPES OF ANTIB...
 
FS P2 COMBO MSTA LAST PUSH past exam papers.
FS P2 COMBO MSTA LAST PUSH past exam papers.FS P2 COMBO MSTA LAST PUSH past exam papers.
FS P2 COMBO MSTA LAST PUSH past exam papers.
 
The Mariana Trench remarkable geological features on Earth.pptx
The Mariana Trench remarkable geological features on Earth.pptxThe Mariana Trench remarkable geological features on Earth.pptx
The Mariana Trench remarkable geological features on Earth.pptx
 
Understanding Partial Differential Equations: Types and Solution Methods
Understanding Partial Differential Equations: Types and Solution MethodsUnderstanding Partial Differential Equations: Types and Solution Methods
Understanding Partial Differential Equations: Types and Solution Methods
 
Clean In Place(CIP).pptx .
Clean In Place(CIP).pptx                 .Clean In Place(CIP).pptx                 .
Clean In Place(CIP).pptx .
 
Efficient spin-up of Earth System Models usingsequence acceleration
Efficient spin-up of Earth System Models usingsequence accelerationEfficient spin-up of Earth System Models usingsequence acceleration
Efficient spin-up of Earth System Models usingsequence acceleration
 
Digital Dentistry.Digital Dentistryvv.pptx
Digital Dentistry.Digital Dentistryvv.pptxDigital Dentistry.Digital Dentistryvv.pptx
Digital Dentistry.Digital Dentistryvv.pptx
 
Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS ESCORT SERVICE In Bhiwan...
Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS  ESCORT SERVICE In Bhiwan...Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS  ESCORT SERVICE In Bhiwan...
Bhiwandi Bhiwandi ❤CALL GIRL 7870993772 ❤CALL GIRLS ESCORT SERVICE In Bhiwan...
 
Selaginella: features, morphology ,anatomy and reproduction.
Selaginella: features, morphology ,anatomy and reproduction.Selaginella: features, morphology ,anatomy and reproduction.
Selaginella: features, morphology ,anatomy and reproduction.
 
module for grade 9 for distance learning
module for grade 9 for distance learningmodule for grade 9 for distance learning
module for grade 9 for distance learning
 
Climate Change Impacts on Terrestrial and Aquatic Ecosystems.pptx
Climate Change Impacts on Terrestrial and Aquatic Ecosystems.pptxClimate Change Impacts on Terrestrial and Aquatic Ecosystems.pptx
Climate Change Impacts on Terrestrial and Aquatic Ecosystems.pptx
 
Site Acceptance Test .
Site Acceptance Test                    .Site Acceptance Test                    .
Site Acceptance Test .
 

I- Function and H -Function Associated With Double Integral

  • 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 16-19 www.iosrjournals.org www.iosrjournals.org 16 | Page I- Function and H -Function Associated With Double Integral Ashok Singh Shekhawat* , Rakeshwar Purohit** And Jyoti Shaktawat*** *Department of Mathematics, Regional college for Education, Research and Technology, Jaipur, Rajasthan (India) **Department of Mathematics and Statistics, University College of Science, Mohan Lal Sukhadia University, Udaipur, Rajasthan (India) *** Department of Mathematics, Kautilya Institute of Technology and Engineering, Jaipur, Rajasthan (India) Abstract: The object of this paper is to discuss certain integral properties of a I -function and H -function, proposed by Inayat-Hussain which contain a certain class of Feynman integrals, the exact partition of a Gaussian model in Statistical Mechanics and several other functions as its particular cases. During the course of finding, we establish certain new double integral relation pertaining to a product involving I function and H function. These double integral relations are unified in nature and act as a key formulae from which we can obtain as their special case, double integral relations concerning a large number of simple special functions. For the sake of illustration, we record here some special cases of our main results which are also new and of interest by themselves. All the result which are established in this paper are basic in nature and are likely to find useful applications in several fields notably electrical network, probability theory and statistical mechanics. Key words. I function, H -function, Hermite polynomials, Laguerre polynomials. I. Introduction The H -function [6] is a new generalization of the well known Fox’s H-function [4]. The H -function pertains the exact partition function of the Gaussian model in statistical mechanics, functions useful in testing hypothesis and several others as its particular cases. The conventional formulation may fail pertaining to the domain of quantum cosmology but Feynman path integrals apply [10,11]. Feynman integral are useful in the study and development of simple and multiple variable hypergeometric series which in turn are useful in statistical mechanics. The I-function defined as          jijiajja jijibjj(b zII[z]            zI ip1,njijian1,jja iq1,mjijibm1,jj{(b nm, iqip  dszs) 2 1 s     where                s)as)b1 s)a1s)b s) jiji ip 1nj jiji iq 1mj1i jj n 1j jj m 1j  …(1.1) The H -function will be defined and represent as given in [1]                dx i2 1 xHx]H i i P1,NjjaN1,jAjj(a Q1,MjjbM1,jj(b NM, QP, NM, QP, …(1.2) where
  • 2. I- Function And H -Function Associated With Double Integral www.iosrjournals.org 17 | Page        jj P 1Nj jj Q 1Mj jjj N 1j jj M 1j ab1 Aa1b …(1.3) which contains fractional powers of some of the gamma functions. Here aj (j = 1,…,P) and bj (j = 1,…,Q) are complex parameters, aj  0 (j = 1,…,P), j  0 (j = 1,…,Q) (not all zero simultaneously and the exponents Aj (j = 1,…,N) and Bj (j = M+1,…,Q) can take on non-integer values. The contour in (1.2) is imaginary axis R() = 0. It is suitably indented in order to avoid the singularities of the gamma functions and to keep those singularities on appropriate side. Again for Aj (j=1,…,N) not an integer, the poles of the gamma function of the numerator in (1.3) are converted to branch points. However, a long as there is no coincidence of pole from any N)1,...,ja1andM)1,...,jb jjjj  pair, the branch cuts can be chosen so that the path of integration can be distorted in the useful manner. For the sake of brevity 0BAT j P 1Nj jj Q 1Mj jj N 1j j M 1j    II. Main Result We will obtain the following result: (A) dydx xy1 yv1 Hvy xy1 x1 I y)x)(1(1 xy1 xy1 y1 vy xy1 x1 NM, QP, 1 0 1 0                                                            vHvIs P1,NjjaN1,jAjja11 1k1Q1,MjBjjbM1,jjb 1NM, 1Q1,P jijiajja jijibjj(b …(2.1) provided that  T 2 1 |Varg0b[R jj Proof. We have                                 s)as)b1 sa1s)b 2 1 v xy1 y)1 Hvy xy1 x1 I jiji ip 1nj jiji iq 1mj1i jj n 1j jj m 1jNM, QP,  .                           dv xy1 y1 ab1 Aa1b i2 1 dsvy xy1 x1 jj P 1Nj jjj Q 1Mj jjj N 1j jj M 1j i i s …(2.2) Multiplying both sides of (2.2) by                          y)x)(1(1 xy1 xy1 y1 y xy1 x1 and integration with respect to x and y between 0 and 1 for both the variable and making a use of a known result [2, p.145],we get the required result (2.1) after a little simplification
  • 3. I- Function And H -Function Associated With Double Integral www.iosrjournals.org 18 | Page (B) dvduv]HI(u)uvv)u NM, QP, 11 00     dzzz)zIs 1 0 jijiajja jijibjj(b            . dzzH P1,NjjaN1,jAjja11 1s1Q1,MjBjjbM1,jjb 1NM, 1Q1,P             …(2.3) provided that  0bR( jj Proof. Using (1.1) and (1.2), we have dsu s)as)b1 sa1s)b 2 1 v]HI(u) s jiji ip 1nj jij iq 1mj1i jj n 1j jj m 1jNM, QP,                   .             dv ab1 Aa1b i2 1 jj P 1Nj jjj Q 1Mj jjj N 1j jj M 1j i i …(2.4) Multiplying both side by 11 uvv)u   and integrating with respect to u and v between 0 and  for both the variable and make a use of a known result [2, p.177], we get the required result. Letting pz ez)   in (2.3), we get the particular case after simplification (c) dvduv]1Hu)]I[(v(1vv)1u)(1f(uv) NM, QP, 11 1 0 1 0    1s 1 0 jijiajja jijibjj(b z)(1f(z)z)(1Is           . dzz)1H P1,NjjaN1,jAjja11 1s1Q1,MjBjjbM1,jjb 1NM, 1Q1,P              …(2.5) provided that R() > 0, R() > 0. Proof. Using equation (1.1) and (1.2), we have dsu)1v s)as)b1 sa1s)b 2 1 v]1Hu)]I[v(1 ss jiji ip 1nj jij iq 1mj1i jj n 1j jj m 1jNM, QP,                   
  • 4. I- Function And H -Function Associated With Double Integral www.iosrjournals.org 19 | Page .             dv)1 ab1 Aa1b i2 1 jj P 1Nj jjj Q 1Mj jjj N 1j jj M 1j i i …(2.6) Multiplying both side of (2.6) by   vv)1u)(1f(uv) 11 and integrating with respect to u and v between 0 and 1 for both the variable and use of result [2, p.243] and by further simplification, we get the result (2.5). Letting f(z) = 1 z  in (2.5), we get the particular result after simplification. Particular Case (i) Taking                 vy xy1 x1 Svy xy1 x1 I m n The result in (2.1), (2.2) and (2.3) reduces to the known result after a slight simplification obtained by Chaurasia and Shekhawat [2]. Acknowledgement. The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and valuable suggestions in the preparation of this paper. References [1]. R.G. Buschman and H.M. Srivastava, The H -function associated with a certain class of Feynman integrals, J. Phys. A: Math. Gen.. 23 (1990), 4707-4710. [2]. V.B.L. Chaurasia and Ashok Singh Shekhawat, Some integral properties of a general class of polynomials associated with Feynman integrals, Bull. Malaysian Math. Sc. Soc. (2) (28) (2) (2005), 183-189. [3]. J. Edewards, A Treatise on integral calculus, Chelsea Pub. Co., 2 (1922). [4]. C. Fox, The G and H-functions as Symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429. [5]. C. Grosche and F. Steiner, Hand Book of Feynman Path integrals, Springer Tracts in Modern Physics Vol.145, Springer-Verlag Berlin Heidelberg, New York, 1998. [6]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals : I. Transformation and reduction formulae, J. Phys. A: Math. Gen. 20 (1987), 4109-4117. [7]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals :II. A generalization of the H- function, J. Phys. A: Math. Gen. 20 (1987), 4119-4128. [8]. Saxena, R.K., On fractional integer operators, Math. Zeitschr, 96 (1967), 288-291. [9]. Saxena, R.K. and Gupta, N., Some Abelian theorems for distributional H-function Transformation, Indian J. Pure Appl. Math. 25(8) (1994), 869-879. [10]. H.M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14 (1972), 1-6. [11]. H.M. Srivastava and N.P. Singh, The integration of certain products of Multivariable H-function with a general class of Polynomials, Rend. Circ. Mat. Palermo 2(32) (1983), 157-187. [12]. C. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23 Fourth edition, Amer. Math. Soc. Providence, Rhode Island (1975).