Some unified integrals associated with Bessel-Struve kernel function

0.13 MB·

by  K. S. Nisar

SOME UNIFIED INTEGRALS ASSOCIATED WITH
BESSEL-STRUVE KERNEL FUNCTION
6
1
K.S.NISAR,P.AGARWAL,ANDS.JAIN
0
2
n Abstract. Inthispaper,wediscussthegeneralizedintegralformulainvolving
a Bessel-StruvekernelfunctionSα(λz),whichexpressedintermsofgeneralized
J Wrightfunctions. Manyinterestingspecialcases alsoobtainedinthisstudy.

1
3

]
A 1. Introduction
C
In 1888 Pincherle studied the integrals involving product of Gamma functions
.
h along vertical lines (see [1, 2, 3]). Latterly, Barnes [4] , Mellin [5] and Cahen [6]
t extended the study and applied some of these integrals in the study of Riemann
a
m zeta function and other Drichlet’s series. The integral formulas involving special
functionshavebeendevelopedbymanyresearchers([7],[8]). In[9]presentedunified
[
integralrepresentationofFoxH-functionsandin[10]hypergeometric F functions.
2 1
1 Recently J. Choi and P. Agarwal[11] obtained two unified integral representations
v
of Bessel functions J (z). Also, many interesting integral formula involving J (z)
6 v v
is given in [7]and [12].
9
4 The Bessel-Struve kernel Sα(λz),λ C, [14] which is unique solution of the
∈
1 initial value problem l u(z) = λ2u(z) with the initial conditions u(0) = 1 and
α
0 ′
u (0)=λΓ(α+1)/√πΓ(α+3/2) is given by
.
2
0 Sα(λz)=jα(iλz) ihα(iλz), z C
− ∀ ∈
6
where j and h are the normalized Bessel and Struve functions
1 α α
: Moreover, the Bessel-Struve kernel is a holomorphic function on C C and it
v ×
can be expanded in a power series in the form
i
X
ar ∞ (λz)nΓ(α+1)Γ((n+1)/2)
(1.1) S (λz)= ,
α
√πn!Γ(n/2+α+1)
nX=0
The generalized Wright hypergeometric function ψ (z) is given by the series
p q
(1.2) ψ (z)= ψ (ai,αi)1,p z = ∞ pi=1Γ(ai+αik)zk,
p q p q(cid:20) (bj,βj)1,q (cid:12)(cid:12) (cid:21) Xk=0 Qqj=1Γ(bj +βjk) k!
(cid:12) Q
where a ,b C, and real α ,β R ((cid:12)i = 1,2,...,p;j = 1,2,...,q). Asymptotic
i j i j
behavior of t∈his function for large∈values of argumentof z C were studied in [21]
∈
2000 Mathematics Subject Classification. Primary05C38,15A15;Secondary 05A15,15A18.
Key words and phrases. Bessel Struve kernel function, generalized Wright functions,Integral
representations.
1
2 K.S.NISAR,P.AGARWAL,ANDS.JAIN
and under the condition
q p
(1.3) β α > 1
j i
− −
Xj=1 Xi=1
was found in the work of [22, 23]. Properties of this generalized Wright function
were investigated in [25], (see also [26, 27]. In particular, it was proved [25] that
ψ (z), z C is an entire function under the condition (1.3).
p q
∈
The generalized hypergeometric function represented as follows [28]:
(1.4) F (αp); z = ∞ Πpj=1(αj)nzn,
p q(cid:20) (βq); (cid:21) nX=0 Πpj=1(βj)n n!
provided p q;p=q+1 and z <1
≤ | |
where(λ) iswellknownPochhammersymboldefinedfor( for λ C)(see[28])
n ∈
1 (n=0)
(1.5) (λ) :=
n (cid:26) λ(λ+1)....(λ+n 1) (n N := 1,2,3.... )
− ∈ { }
Γ(λ+n)
(1.6) (λ)n = Γ(λ) λ∈C\Z0− .
(cid:0) (cid:1)
where Z0− is the set of nonpositive integers.
If we put α = ... = α = β = .... = β in (1.2),then (1.4) is a special case of
1 p 1 q
the generalized Wright function:
(1.7) ψ (z)= ψ (α1,1),...,(αp,1); z = pj=1Γ(αj) F α1,...,αp; z
p q p q(cid:20) (β1,1),...,(βq,1); (cid:21) Qqj=1Γ(βj) p q(cid:20) β1,...,βq; (cid:21)
Q
For the present investigation, we need the following result of Oberbettinger [30]
(1.8) ∞xµ−1 x+a+ x2+2ax −λdx=2λa−λ a µ Γ(2µ)Γ(λ−µ)
Z0 (cid:16) p (cid:17) (cid:16)2(cid:17) Γ(1+λ+µ)
provided 0<Re(µ)<Re(λ)
Motivated by the work of [8] , here we present the integral formulas of Bessel-
Struve Kernel function of first kind S (λz) ,λ C,which expressed interns of
α
∈
generalized Wright or generalized hypergeometric functions.
2. Main results
Two generalized integral formulas established here, which expressed in terms of
generalized(Wright) hypergeometric functions (1.7) by inserting the Bessel-Struve
kernel function of the first kind (1.1) with the suitable argument in the integrand
of (1.8)
Theorem 1. For λ,µ,ν,γ C,and x > 0, R(λ) > R(µ) > 0, then the following
∈
integral formula holds true:
SOME UNIFIED INTEGRALS ASSOCIATED WITH BESSEL-STRUVE KERNEL FUNCTION 3
∞xµ 1 x+a+ x2+2ax −λS γy dx
− α
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
21 µaµ λΓ(α+1)Γ(2µ)
− −
=
√π
1,1 ,(λ+1,1),(λ µ,1); γy
(2.1) ×3Ψ2(cid:20) (cid:0)2 (2λ(cid:1),1),(1+λ+µ,−1); a (cid:21)
Proof. ConsidertheseriesrepresentationofS γy andapplying(1.8).
α x+a+√x2+2ax
(cid:16) (cid:17)
Byinterchangingtheorderofintegrationandsummation,whichverifiedbyuniform
convergence of the involved series under the given conditions, we get
∞xµ 1 x+a+ x2+2ax −λS γy dx
− α
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
n
γy Γ(α+1)Γ n+1
= Z0∞xµ−1(cid:16)x+a+px2+2ax(cid:17)−λnX∞=0(cid:16)x+a+√√x2π+Γ2axn2(cid:17)+α+1 n! (cid:0) 2 (cid:1)dx
= ∞ (γy)nΓ(α+1)Γ n+21 ∞xµ 1 x+a+ x2(cid:0)+2ax −(λ(cid:1)+n)dx
nX=0 √πΓ n2 +α+1(cid:0) n! (cid:1)Z0 − (cid:16) p (cid:17)
(cid:0) (cid:1)
in view of the conditions give in Theorem 1and applying the integral formula
(1.8) ,we obtain the following integral representation:
∞xµ 1 x+a+ x2+2ax −λS γy dx
− α
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
∞ n+1 Γ(λ+n+1)
= 21−µa−λ+µΓ(α+1)Γ(2µ)π−1/2 Γ
(cid:18) 2 (cid:19) Γ(λ+n)
nX=0
Γ(λ+n µ) γnyn
−
×Γ(λ+µ+n+1) n!an
which,upon using (1.7) ,yeilds (2.1). This completes proof of theorem 1 (cid:3)
Theorem 2. For λ,µ,ν,γ C with 0 < R(µ) < R(λ+ν) and x > 0. The
∈
following integral formula hold true:
∞xµ 1 x+a+ x2+2ax −λS γxy dx
− α
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
= 21+µaµ−λΓ(α+1)Γ(λ−µ) Ψ 21,12 ,(2µ,2),(λ+1,1); γy
√πΓ(1+λ+µ) 3 2(cid:20) (cid:0) (λ(cid:1),1), α+1,1 ; (cid:21)
2
(cid:0) (cid:1)
4 K.S.NISAR,P.AGARWAL,ANDS.JAIN
Proof. Interchanging the order of integration and summation and the series repre-
sentation of Bessel Struve kerenel function, we get
∞xµ−1 x+a+ x2+2ax −λSα γxy dx
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
= ∞xµ 1 x+a+ x2+2ax −λ
−
Z0 (cid:16) p (cid:17)
∞ γxy n Γ(α+1)Γ n+1
2 dx
× (cid:18)x+a+√x2+2ax(cid:19) √πΓ n +α(cid:0)+1 (cid:1)n!
nX=0 2
(cid:0) (cid:1)
= ∞ γnynΓ(α+1)Γ n+21 ∞xµ+n 1 x+a+ x2+2ax −(λ+n)dx
nX=0 √πΓ n2 +α+(cid:0)1 (cid:1)Z0 − (cid:16) p (cid:17)
(cid:0) (cid:1)
inviewoftheconditiongiveintheorem1,wecanapplytheintegralformula(1.8)
and obtain the following integral representation:
∞xµ 1 x+a+ x2+2ax −λS γxy dx
− α
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
21+µaµ−λΓ(α+1)Γ(λ µ) ∞ n+1 Γ(λ+n+1)Γ(2µ+2n)γnyn
= − Γ
√πΓ(1+λ+µ) (cid:18) 2 (cid:19) Γ(λ+n)Γ n +α+1 n!an
nX=0 2
(cid:0) (cid:1)
which gives the desired result. (cid:3)
2.1. Representation of Bessel Struve kernel function in terms of expo-
nential function. In this subsection we represent the Bessel Struve function in
terms ofexponentialfunction. Also, we derive the MarichevSaigoMaeda operator
representation of special cases. The representation Bessel Struve Kernel function
interms of exponential function as:
(2.2) S−1 (x)=ex,
2
1+ex
(2.3) S1 (x)= − .
2 x
Now, we give the the following corollaries:
Corollary1. Forλ,µ Cwith0<R(µ)<R(λ)andx>0.Thefollowing integral
∈
formula holds true
(2.4) ∞xµ−1 x+a+ x2+2ax −λe(cid:18)x+a+√yx2+2ax(cid:19)dx
Z0 (cid:16) p (cid:17)
(λ+1,1),(λ µ,1); y
= 21−µaµ−λΓ(2µ) 2Ψ2(cid:20) (λ,1),(1+λ− µ); a(cid:21)
−
Proof. As same as in theorem 1 and theorem 2 , using the formula (1.8) and (2.2),
one can easily reach the result (cid:3)
SOME UNIFIED INTEGRALS ASSOCIATED WITH BESSEL-STRUVE KERNEL FUNCTION 5
Corollary 2. Let the conditions given in Corollary 1 satisfied. Then the following
integral formula holds true
∞xµ−1 x+a+ x2+2ax −λe(cid:18)x+a+√yx2+2ax(cid:19)dx
Z0 (cid:16) p (cid:17)
21−µaµ−λΓ(2µ)Γ(λ+1)Γ(λ µ) λ+1,λ µ; y
= = Γ(λ)Γ(1+λ µ) − 2F2(cid:20) λ,1+λ−µ; a(cid:21)
− −
Proof. In the view of equations (1.4) , (1.5) and (2.4) , we obtain the required
result. (cid:3)
Corollary 3. For λ,µ, C with 0 < R(µ) < R(λ) and x > 0.The following
∈
integral formula holds true
∞xµ−1 x+a+ x2+2ax −λe(cid:18)x+a+√yx2+2ax−1(cid:19)dx
Z0 (cid:16) p (cid:17)
1,1 ,(λ+1,1),(λ µ,1); y
= 2−µaµ−λΓ(2µ) 3Ψ3(cid:20) (cid:0)12,23(cid:1),(λ,1),(1+λ−+µ,1); a(cid:21)
2 2
(cid:0) (cid:1)
2.2. RelationbetweenBesselStruvekernelfunctionandBesselandStruve
function of first kind. In this subsection we show the relation between S (x)
α
andBesselfunction I (x)andStruvefunction L (x)by choosingparticularvalues
v v
of α
(2.5) S (x)=I (x)+L (x),
0 0 0
2I (x)+L (x)
1 1
(2.6) S (x)= ,
1
x
In the light of above relations ,we have the following theorems:
Theorem 3. For λ,µ C with 0 < R(µ) < R(λ) and x > 0.Then the following
∈
integral formula holds true:
∞xµ 1 x+a+ x2+2ax −λ
−
Z0 (cid:16) p (cid:17)
y y
I +L dx
0 0
×(cid:20) (cid:18)x+a+√x2+2ax(cid:19) (cid:18)x+a+√x2+2ax(cid:19)(cid:21)
1,1 ,(λ+1,1),(λ µ,1); y
= 21−µaµ−λπ−1/2Γ(2µ) 3Ψ3(cid:20) (cid:0)12,12(cid:1),(λ,1),(1+λ+−µ,1); a(cid:21)
2
(cid:0) (cid:1)
Proof. Consider the relation given in (2.5) and applying (1.8). By interchanging
the order of integration and summation,which verified by uniform convergence of
6 K.S.NISAR,P.AGARWAL,ANDS.JAIN
the involved series under the given conditions, we get
∞xµ 1 x+a+ x2+2ax −λ
−
Z0 (cid:16) p (cid:17)
y y
I +L dx
0 0
×(cid:20) (cid:18)x+a+√x2+2ax(cid:19) (cid:18)x+a+√x2+2ax(cid:19)(cid:21)
= ∞xµ 1 x+a+ x2+2ax −λ ∞ y n Γ n+21 dx
Z0 − (cid:16) p (cid:17) nX=0(cid:18)x+a+√x2+2ax(cid:19) √πn!(cid:0)Γ n2 (cid:1)+1
(cid:0) (cid:1)
= ∞ Γ n+21 yn ∞xµ 1 x+a+ x2+2ax −(λ+n)dx
nX=0√πn(cid:0)!Γ n2(cid:1)+1 Z0 − (cid:16) p (cid:17)
(cid:0) (cid:1)
∞ Γ n+1 yn Γ(λ+n+1) Γ(λ+n µ)
= 21−µaµ−λπ−1/2Γ(2µ) n!Γ(cid:0)n2+(cid:1)1 an Γ(λ+n) Γ(λ+n+−1+µ)
nX=0 2
(cid:0) (cid:1)
which gives the desired result. (cid:3)
Theorem 4. The following integral formula holds true with λ,µ, C with 0 <
R(µ)<R(λ) and x>0. ∈
∞xµ−1 x+a+ x2+2ax −λ
Z0 (cid:16) p (cid:17)
y y
2I +L dx
1 1
×(cid:20) (cid:18)x+a+√x2+2ax(cid:19) (cid:18)x+a+√x2+2ax(cid:19)(cid:21)
1,1 ,(λ µ,1); y
= 21−µaµ−λπ−1/2Γ(2µ) 2Ψ2(cid:20) 2(cid:0),21 2,(cid:1)(1+λ−+µ,1); a(cid:21)
2
(cid:0) (cid:1)
Proof. ConsidertheseriesrepresentationofS y andapplying(1.8)and
1 x+a+√x2+2ax
(cid:16) (cid:17)
interchanging the order of integration and summation, we get
∞xµ 1 x+a+ x2+2ax −λ
−
Z0 (cid:16) p (cid:17)
y y
2I +L dx
1 1
×(cid:20) (cid:18)x+a+√x2+2ax(cid:19) (cid:18)x+a+√x2+2ax(cid:19)(cid:21)
= ∞xµ 1 x+a+ x2+2ax −λS y dx
− 1
Z0 (cid:16) p (cid:17) (cid:18)x+a+√x2+2ax(cid:19)
= ∞ Γ n+21 yn ∞xµ 1 x+a+ x2+2ax −(λ+n)dx
nX=0√πn(cid:0)!Γ n2(cid:1)+2 Z0 − (cid:16) p (cid:17)
(cid:0) (cid:1)
∞ Γ n+1 Γ(λ+n µ) yn
= 21 µaµ λπ 1/2Γ(2µ) 2 −
− − − Γ n(cid:0)+2 (cid:1)Γ(1+λ+n+µ)ann!
nX=0 2
(cid:0) (cid:1)
which gives the required result. (cid:3)
Conclusion
ThegeneralizedintegralformulainvolvingBessel-StruvekernelfunctionS (λz),
α
which expressed in terms of generalized Wright functions are given in this paper.
Alsotherelationbetweenexponentialfunction,BesselfunctionandStruvefunction
with Bessel-Struve kernel function is also discussed with particular cases.
SOME UNIFIED INTEGRALS ASSOCIATED WITH BESSEL-STRUVE KERNEL FUNCTION 7
References
[1] S.Pincherle,SullefunzioniipergeometrichegeneralizzateNotaI,AttidellaRealeAccademia
dei Lincei. Rendiconti dellaClassedi Scienze Fisiche, Matematiche e Naturali, Serie4, Vol.
4(1888), 694–700
[2] S, Pincherle: Sulle funzioni ipergeometriche generalizzate. Nota I-II. In: Opere Scelte-I,
UnioneMatematicaItaliana,Ed.Cremonese,Rome(1954), 223-239.
[3] Mainardi,F,Pagnini,G,Pincherle,S:ThepioneeroftheMellin-Barnesintegrals.J.Comput.
Appl.Math.153,(2003),331-342.
[4] Barnes,EW:Anewdevelopmentofthetheoryofthehypergeometricfunctions.Proc.Lond.
Math.Soc.6,141-177,(1908)
[5] Mellin,HM:AbrißeinereinheitlichentheoriederGammaundderhypergeometrischenfunk-
tionen.Math.Ann.68,305-337(1910)
[6] Cahen,E:SurlafonctionξσdeRiemannetsurdesfonctionsanalogues.Ann.Sci.E´c.Norm.
Super.11,75-164,(1894)
[7] Brychkov,YA:Handbookofspecialfunctions: derivatives,integrals,seriesandotherformu-
las.CRCPress,BocaRaton,(2008)
[8] Choi, J, Hasanov, A, Srivastava, HM, Turaev, M: Integral representations for Srivastava’s
triplehypergeometricfunctions.Taiwan.J.Math.15,2751-2762(2011)
[9] Garg, M, Mittal, S: On a new unified integral. Proc. Indian Acad. Sci. Math. Sci. 114(2),
99-101(2003)
[10] Ali,S:Onsomenewunifiedintegrals.Adv.Comput.Math.Appl.1(3), 151-153(2012)
[11] JunesangChoi,PraveenAgarwal,Certainunifiedintegralsassociated withBesselfunctions,
BoundaryValueProblems2013,2013:95
[12] Watson,GN:ATreatiseontheTheoryofBesselFunctions.CambridgeMathematicalLibrary
Edition.CamdridgeUniversityPress,Camdridge(1995). Reprinted(1996)
[13] H.Struve,BeitragzurTheoriederDiffractionanFernro¨hren,Ann.PhysikChemie,17(1882),
1008–1016.
[14] A. Gasmi and M. Sifi, The Bessel-Struve intertwinning operator on C and mean-periodic
functions,IJMMS2004:59, 3171–3185
[15] A. R. Ahmadi and S. E. Widnall, Unsteady lifting-line theory as a singular-perturbation
problem.J.FluidMech.153(1985), 59-81.
[16] M.H.Hirata,Flownearthebowofasteadilyturningship,J.FluidMech.71(1975), no.2,
283-291.
[17] D. C. Shaw, Perturbational results for di raction of water-waves by nearly-vertical barriers.
IMA,J.Appl.Math.34(1985), no.1,99-117.
[18] J. Shao, P. Hanggi, Decoherent dynamics of a two-level system coupled to a sea of spins.
Phys.Rev.Lett.81(1998), no.26,5710-5713.
[19] J. Shao, P. Hanggi, Decoherent dynamics of a two-level system coupled to a sea of spins.
Phys.Rev.Lett.81(1998), no.26,5710-5713.
[20] A.Erd´elyi,W.Magnus,F.OberhettingerandF.G.Tricomi,Highertranscendentalfunctions.
Vols. I, II,McGraw-HillBookCompany, Inc.,NewYork,1953. NewYork,Toronto, London,
1953.
[21] C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London.
Math.Soc.27(1928), no.4,389-400.
[22] E.M.Wright,TheasymptoticexpansionofintegralfunctionsdefinedbyTaylorseries,Philos.
Trans.Roy.Soc.London, Ser.A.238(1940), 423–451.
[23] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc.
LondonMath.Soc.(2)46(1940), 389-408.
[24] A.A.KilbasandM.Saigo,H-transforms, Chapman&Hall/CRC,BocaRaton,FL,2004.
[25] A. A. Kilbas, M. Saigo and J. J. Trujillo, On the generalized Wright function, Fract. Calc.
Appl.Anal.5(2002), no.4,437–460.
[26] A. A. Kilbas and N. Sebastian, Generalized fractional integration of Bessel function of the
firstkind,IntegralTransformsSpec.Funct.19(2008), no.11-12,869–883.
[27] A.A.KilbasandN.Sebastian,FractionalintegrationoftheproductofBesselfunctionofthe
firstkind,Fract.Calc.Appl.Anal.13(2010), no.2,159–175.
[28] E.D.Rainville,Special functions,Macmillan,NewYork,1960.
8 K.S.NISAR,P.AGARWAL,ANDS.JAIN
[29] Saiful.R.Mondal,UnifiedIntegrals associated withgeneralizedBesselfunctions andStruve
functions,Eng.Math.Let,10,2015
[30] Oberhettinger,F:TablesofMellinTransforms.Springer,NewYork(1974)
Department of Mathematics, College of Arts and Science-Wadi Addwasir, Prince
SattambinAbdulaziz University,SaudiArabia
E-mail address: [email protected]
Department of Mathematics, Anand International College of Engineering, Jaipur
303012,Rajasthan,India.
Department of Mathematics, Poornima College of Engineering,Jaipur 302002, Ra-
jasthan,India.