Certain unified integral formulas involving the generalized modified k-bessel function of first kind

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by  K. S. Nisar

CERTAIN UNIFIED INTEGRAL FORMULAS INVOLVING THE
GENERALIZED MODIFIED K-BESSEL FUNCTION OF FIRST
6 KIND
1
0
2 K.S.NISARANDS.R.MONDAL

n
a Abstract. Generalized integral formulas involving the generalized modified
J k-Bessel function Jc,γ,λ(z) of first kind are expressed in terms generalized
k,ν
5 k−Wright functions .Some interesting special cases of the main results are
2 alsodiscussed

]
A
C 1. Introduction
.
h
t The integral formula involving various special functions have been studied by
a
m many researchers ([7],[8]). In 1888 Pincherle gave the integrals involving product
of Gamma functions along vertical lines (see [1, 2, 3]). Barnes [4] , Mellin [5] and
[
Cahen [6] extended some of these integrals in the study of Riemann zeta function

1 and other Drichlet’s series. The integral representation of Fox H-functions and
v
hypergeometric F functionsstudiedby[9]and[10]respectively. Also,theintegral
7 2 1
representationofBesselfunctions aregiveninmany recentworks(see [11], [12], [7]
8
4 and [15]).
6 let k R;α,λ,γ,υ C;Re(λ) >0,Re(υ)>0,the k-Bessel function of the first
0 kind defi∈ned by the fol∈lowing series [13] :
.
1
0 ∞ (γ) ( 1)n(z/2)n
6 (1.1) J(γ),(λ)(z)= n,k −
1 k,ν nX=0Γk(λn+υ+1) (n!)2
:
v where (γ) is the k Pochhammer symbol [25] is defined as:
i n,k −
X
r (1.2) (x) =x(x+k)(x+2k)...(x+(n 1)k),γ C,k R and n N
a n,k − ∈ ∈ ∈
and Γ (z) is the k gamma function, k be the positive real number, defined by
k
−
(see [25])
∞
(1.3) Γk(z)= e−tkktz−1dt,Re(z)>0
Z
0
Clearly, for k =1, Γ (z) reduces to the classical Γ(z) function.
k
In this paper, we introduce a new generalization of k Bessel function called
−
generalized modified k Bessel function and is defined as:
−
1991 Mathematics Subject Classification. Primary 33B20; 33C20; 26A33; secondary 33B15;
33C05.
Key words and phrases. k-Bessel function; Gamma function; hypergeometric function 2F1;
generalized hypergeometric function pFq; generalized (Wright) hypergeometric functions pΨq;
Oberhettinger’sintegralformula;
1
2 K.S.NISARANDS.R.MONDAL
let k R;α,λ ,γ,υ,c,b C;Re(λ ) > 0,Re(υ) > 0,the generalized modified
1 1
∈ ∈
k Bessel function of the first kind given by the following series
−
∞ (c)n(γ) (z/2)υ+2n
(1.4) Jc,γ,λ(z)= n,k
k,ν nX=0Γk λ1n+υ+ b+21 (n!)2
(cid:0) (cid:1)
Theaimofthispaperistoestablishtwogeneralizedintegralformulas,whichare
expressed in terms of generalized k Wright functions, by inserting newly general-
−
ized modified k-Bessel function.
The generalized Wright hypergeometric function ψ (z) is given by the series
p q
(1.5) 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 [17]
∈
and under the condition
q p
(1.6) β α > 1
j i
− −
Xj=1 Xi=1
was found in the work of [18, 19]. Properties of this generalized Wright function
were investigated in [21], (see also [22, 23]. In particular, it was proved [21] that
ψ (z), z C is an entire function under the condition (1.6).
p q
∈
The generalized hypergeometric function represented as follows [24]:
(1.7) 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[24])
n ∈
1 (n=0)
(1.8) (λ) :=
n (cid:26) λ(λ+1)....(λ+n 1) (n N := 1,2,3.... )
− ∈ { }
Γ(λ+n)
−
(1.9) (λ) = λ C Z .
n Γ(λ) ∈ \ 0
(cid:0) (cid:1)
−
where Z is the set of non-positive integers.
0
If we put α = ... = α = β = .... = β in (1.5),then (1.7) is a special case of
1 p 1 q
the generalized Wright function:
p
(1.10) ψ (z)= ψ (α1,1),...,(αp,1); z = j=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 [14]
∞ −λ a µ Γ(2µ)Γ(λ µ)
(1.11) xµ−1 x+a+ x2+2ax dx=2λa−λ −
Z0 (cid:16) p (cid:17) (cid:16)2(cid:17) Γ(1+λ+µ)
provided 0<Re(µ)<Re(λ)
SOME UNIFIED INTEGRALS OF GENERALIZED K-BESSEL FUNCTION 3
Also,weneedthefollowingrelationofΓ withtheclassicalgammaEulerfunction
k
(see [26]:
(1.12) Γ (z+k)=zΓ (z)
k k
(1.13) Γk(z)=kkz−1Γ z
k
(cid:16) (cid:17)
(1.14) Γ (k)=1
k
2. Main results
Two generalized integral formulas established here, which expressed in terms
of generalized k Wright functions (1.10) by inserting the generalized modified k-
−
Bessel function of the first kind (1.4) with the suitable argument in the integrand
of (1.11)
Theorem 1. For λ,µ,ν,c,λ C, Re(λ+ν+2)> Re(µ) > 0 and x > 0. Then
1
∈
the following formula holds true:
∞ −λ y
xµ−1 x+a+ x2+2ax Jc,γ,λ1 dx
Z0 (cid:16) p (cid:17) k,υ (cid:18)x+a+√x2+2ax(cid:19)
= 21−ν−µaµ−λ−υyυk−2µΓ(2µ)
(λ+υ+k,2),(k(ν+λ µ),2k); cy2
(2.1) ×k,2Ψ3(cid:20) ν+ b+21,λ1 ,(k(1+λ+υ+µ−),2k),(λ+ν,2); 4a2(cid:21)
where Ψ denot(cid:0)e the k-Fox-(cid:1)Wright function [26]
k,2 3
Proof. By applying (1.4) to the LHS of (2.1) and interchanging the order of inte-
gration and summation, which is verified by uniform convergence of the involved
series under the given conditions , we obtain
∞ −λ y
xµ−1 x+a+ x2+2ax Jc,γ,λ1 dx
Z0 (cid:16) p (cid:17) k,υ (cid:18)x+a+√x2+2ax(cid:19)
∞ −λ
= xµ−1 x+a+ x2+2ax
Z0 (cid:16) p (cid:17)
∞ (c)n(γ) y υ+2n −(υ+2n)
n,k 2 x+a+ x2+2ax dx
×nX=0Γk λ1n+υ+ b+21 (cid:0)((cid:1)n!)2 (cid:16) p (cid:17)
∞ ((cid:0)c)n(γ) (cid:1)y υ+2n ∞ −(λ+ν+2n)
= n,k 2 xµ−1 x+a+ x2+2ax dx
nX=0Γk λ1n+υ+ b+21 (cid:0)((cid:1)n!)2 Z0 (cid:16) p (cid:17)
In view of(cid:0)the conditions(cid:1)given in Theorem 1, since R(λ+ν) > R(µ) > 0
k N :=N 0 .
0
∈ ∪{ }
Applying (1.11) to the integrand of (2.1) and obtain the following expression:
∞ (c)n(γ) y υ+2n a µ
= n,k 2 2(λ+ν+2n)a−(λ+ν+2n)
nX=0Γk(λ1n+υ+1)(cid:0)((cid:1)n!)2 (cid:16)2(cid:17)
Γ(2µ)Γ(λ+ν+2n µ)
−
× Γ(1+λ+ν+µ+2n)
4 K.S.NISARANDS.R.MONDAL
By making the use of the relation (1.12), we obtain
= 21−ν−µaµ−λ−υyυk−2µΓ(2µ)
∞
Γ (λ+ν+k+2n)Γ (λk+νk µk+2kn)
k k
−
× Γ (λ+ν2n)Γ λ n+υ+ b+1 Γ (k+λk+νk+µk+2kn)
Xk−0 k k 1 2 k
(c)n y2 n (cid:0) (cid:1)
×(n!)2 (cid:18)4a2(cid:19)
which is the desired result. (cid:3)
Corollary 1. Let the conditions of Theorem 1 be satisfied and let k =λ =1 and
1
c= c in (2.1). Then the following integral formula holds:
−
∞ −λ y
xµ−1 x+a+ x2+2ax Jc,γ,λ1 dx
Z0 (cid:16) p (cid:17) k,υ (cid:18)x+a+√x2+2ax(cid:19)
= 21−ν−µaµ−λ−υyυΓ(2µ)
(1+λ+υ,2),(ν+λ µ,2); cy2
×2Ψ3(cid:20) ν+ b+21,λ1 ,(1+λ+υ+µ−,2),(λ+ν,2); −4a2 (cid:21)
(cid:0) (cid:1)
which is the result given by [12] .
Corollary 2. Setting b = c = 1 in (2.1) with some appropriate parameter re-
placements, we get the integral formula of Bessel function J (z) given by Choi and
υ
Agarwal [11].
Theorem 2. For λ ,µ,ν,c,b C,0 < R(µ+ν+2) < R(λ+ν+2) and x > 0,
1
∈
then the following integral formula holds true:
∞ −λ xy
xµ−1 x+a+ x2+2ax Jc,γ,λ1 dx
Z0 (cid:16) p (cid:17) k,υ (cid:18)x+a+√x2+2ax(cid:19)
= 21−2ν−µyνaµ−λk1+λ−µΓ(λ µ)
−
(k(2µ+2ν),4k),(ν+λ+k,2); cy2
(2.2) Ψ
×k,2 3(cid:20) (ν+1,λ1),(ν+λ,2),(k(1+λ+µ+2ν),4k); 4 (cid:21)
where Ψ denote the k-Fox-Wright function [26]
k,2 3
Proof. By applying (1.4) to the LHS of (2.2) and interchanging the order of inte-
gration and summation, which is verified by uniform convergence of the involved
series under the given conditions , we obtain
SOME UNIFIED INTEGRALS OF GENERALIZED K-BESSEL FUNCTION 5
∞ −λ xy
xµ−1 x+a+ x2+2ax Jc,γ,λ1 dx
Z0 (cid:16) p (cid:17) k,υ (cid:18)x+a+√x2+2ax(cid:19)
∞ −λ
= xµ−1 x+a+ x2+2ax
Z0 (cid:16) p (cid:17)
∞ (c)n(γ) xy υ+2n −(υ+2n)
n,k 2 x+a+ x2+2ax dx
×nX=0Γk λ1n+υ+ b+21 (cid:0) (n(cid:1)!)2 (cid:16) p (cid:17)
∞ ((cid:0)c)n(γ) (cid:1)y υ+2n
= n,k 2
nX=0Γk λ1n+υ+ b+21 (cid:0)((cid:1)n!)2
∞ (cid:0) (cid:1) −(λ+υ+2n)
x(µ+υ+2n)−1 x+a+ x2+2ax dx
×Z0 (cid:16) p (cid:17)
Applying (1.11) to the integrand of (2.2) ,we obtain the following expression:
∞ (c)n(γ) y υ+2n a µ+υ+2n
= n,k 2 2(λ+υ+2n)a−(λ+υ+2n)
nX=0Γk λ1n+υ+ b+21 (cid:0)((cid:1)n!)2 (cid:16)2(cid:17)
(cid:0) (cid:1)
Γ(2µ+2ν+4n)Γ(λ µ)
−
× Γ(1+λ+2ν+µ+4n)
By making the use of (1.12), we obtain
= 21−2ν−µyνaµ−λkλ−µ+1Γ(λ µ)
−
∞ (γ)n,kΓk(λ+ν+k+2n)Γk(2µk+2νk+4nk) cy2 n
× (n!)2Γ λ n+υ+ b+1 Γ (λ+ν+2n)Γ ((1+λ+µ+2υ+4n)k)(cid:18) 4 (cid:19)
kX−0 k 1 2 k k
which is the desire(cid:0)d result. (cid:1) (cid:3)
Corollary 3. Let the conditions given in Theorem 2 satisfied and set k = λ =
1
1,and c= c Theorem 2 reduces to
−
∞ −λ xy
xµ−1 x+a+ x2+2ax Jc,γ,1 dx
Z0 (cid:16) p (cid:17) 1,υ (cid:18)x+a+√x2+2ax(cid:19)
= 21−2ν−µyνaµ−λΓ(λ µ)
−
(2µ+2ν,4),(ν+λ+1,2); cy2
Ψ
×2 3(cid:20) ν+ b+21,λ1 ,(ν+λ,2),(1+λ+µ+2ν,4); 4 (cid:21)
(cid:0) (cid:1)
which is the result given by [12] .
Corollary 4. setting b=c=1 in (2.1) with some appropriate parameter replace-
ments, we get the integral formula of Bessel function J (z) given by Choi and
υ
Agarwal [11].
Conclusion
The integral formulas for generalized modified k Bessel function of first kind is
−
derivedandtheresultsareexpressedintermofgeneralizedk Wrightfunction.Some
−
of interesting special cases also derivedfrom the main results. Using some suitable
parametric replacement, theorems 1 and 2 gives the unified integralrepresentation
of generalized Bessel function, if c = 1 and integral representation of modified
−
Bessel function by , if c=1 .
6 K.S.NISARANDS.R.MONDAL
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SOME UNIFIED INTEGRALS OF GENERALIZED K-BESSEL FUNCTION 7
DepartmentofMathematics,CollegeofArts&Science-WadiAddwaser,PrinceSat-
tambinAbdulaziz University,SaudiArabia
E-mail address: [e-mail]
Department of Mathematics, College of Science-Al Ahsa, King Faisal University,
SaudiArabia
E-mail address: [e-mail]