[11] obtained fascinating results by applying the integral formulas linked with the product of Bessel function of first kind and Jacobi polynomials

(1)

https://doi.org/10.5831/HMJ.2019.41.1.135

EVALUATION OF INTEGRAL FORMULAS ASSOCIATED WITH THE PRODUCT OF GENERALIZED BESSEL FUNCTION WITH

ORTHOGONAL POLYNOMIALS

Nabiullah Khan, Raghib Nadeem, Talha Usman^∗ and Abdul Hakim Khan

Abstract. In the last decades, various integral formulas associated with Bessel functions of different kinds as well as Bessel functions themselves, have been studied and a noteworthy amount of work can be found in the literature. Following up, we present two definite integral formulas involving the product of generalized Bessel function associated with orthogonal polynomials. Also, some intriguing special cases of our main results have been discussed.

1. Introduction

Various integral operators comprising assorted kind of special functions have been a dynamic research area through many authors(see, [7],[8],[9],[15],[16],[12],[13]) in the recent years and they have a large num- ber of application in the diverse field of physical sciences, e.g., plasma physics, radio physics, neutrons physics etc.

In 2016, Khan et al. [11] obtained fascinating results by applying the integral formulas linked with the product of Bessel function of first kind and Jacobi polynomials. Very recently, Ghayasuddin et al [10] established two double integrals comprising the product of Bessel function with Jacobi and Laguerre polynomials.

Received July 15, 2018. Revised January 18, 2019. Accepted January 21, 2019.

2010 Mathematics Subject Classification. 33C05, 33C45, 33C47, 33C90.

Key words and phrases. Generalized Bessel function, Orthogonal polynomials, Integral representation.

*Corresponding author

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Motivated by the aforementioned research and success of the application of integral formulas, in the present article, we evaluate a new type of definite integral formula involving the multiplication of generalized Bessel function with Jacobi polynomial and further written in terms of Srivas- tava and Daust function. Also, some other fascinating integrals whose integrands are associated, in particular, Bessel (sine and cosine) function with ultraspherical polynomials, Gegenbauer polynomials, Tchebicheff polynomials and Legendre polynomials are also derived as a particular cases in our main results.

The generalized Bessel function is the solution of the following differ- ential equation:

(1) y²w⁰⁰(y) + byw⁰(y) + [cy²− σ²+ (1 − b)σ]w(y) = 0

where b, c, σ ∈ C with <(σ) > −1. A particular solution of the equation (1) known as generalized Bessel function w_σ,c^b (y) of first kind of order σ and is defined as (see [17]):

(2) w_σ,c^b (y) =

∞

X

r=0

(−1)^rc^r(y/2)^σ+2r

r!Γ(σ + r + ^1+b₂ ), (∀y ∈ C\{0}),

where C represents the set of complex numbers and Γ(.) is the classical gamma function.

By assuming some particular values of the parameters, equation (1) reduces to

Table 1

S. No. b c σ w_σ,c^b (y)

1 1 1 σ Bessel function of first kind Jσ(y) 2 1 -1 σ Modified Bessel function Iσ(y) 3 b c² −^b₂ w₋^b b

2,c²(y) =

2 y

^b₂

coscy√ π

4 b −c² −^b₂ w^b

−₂^b,−c²(y) =

2 y

^b₂

Coshcy√ π

5 b c² 1 −^b₂ w^b

1−^b₂,c²(y) =

2 y

^b₂

sincy√ π

6 b −c² 1 −^b₂ w^b

1−₂^b,−c²(y) =

2 y

₂^b

sinhcy√ π

(3)

The well-known definition of classical Jacobi polynomials P_r^(α,β)(y) are given underneath (see, [2],[3]):

(3) P_r^(α,β)(y) = (1 + α)_r r! ²F1





−r, 1 + α + β + r;

1 + α;

1 − y 2



, or, equivalently,

(4) P_r^(α,β)(y) =

r

X

l=0

(1 + α)r(1 + α + β)r+l

l! (r − l)! (1 + α)_l(1 + α + β)r

y − 1 2

l

. From equation (3), we have

(5) P_r^(α,β)(1) = (1 + α)_r

r! , where Pr^(α,β)(y) is a polynomial of degree r.

On taking α = β in equation (3), the polynomials Pr^(α,α)(y) is called the ultraspherical polynomials

(6) P_r^(α,α)(y) =

r

X

l=0

(1 + α)r(1 + 2α)_r+l l! (r − l)! (1 + α)_l(1 + 2α)_r

y − 1 2

l

.

and further on setting α = β = δ − ¹₂ in (3), the result reduces to:

(7) P^(δ−

1 2,δ−¹₂)

r (y) = (δ +¹₂)r

(2δ)r

C_r^δ(y), where C_r^δ(y) is the Gegenbauer polynomials (see[2],[3]).

Furthermore, α = β = −¹₂ and α = β = ¹₂ respectively in equation (3), we get

(8) P⁽⁻

1 2,−¹₂)

r (y) =

1 2

r

r! Tn(y) and

(9) P⁽

1 2,¹₂) r (y) =

3 2

r

(r + 1)!U_r(z),

where Tr(y) and Ur(y) are Tchebicheff polynomials of first and second kind respectively (see [2],[3]).

For α = β = 0 in equation (3), we have

(10) P_r^(0,0)(y) = Pr(y),

where P_r(y) is the Legendre polynomials (see [2],[3]).

(4)

The generalized Laguerre polynomials L^(α)_r (y) are defined by (see [2],[3]):

(11) L^(α)_r (y) = (1 + α)r

r! ¹F1(−r; 1 + α; y).

For α = 0 the above polynomial is called the Laguerre or simple Laguerre polynomial.

From (11), it follows that:

(12) L^(α)_r (y) =

r

X

l=0

(−1)^l(1 + α)ry^l l!(r − l)!(1 + α)_l. It is also well known that:

(13) L⁻

1

r 2(y²) = H_2r(y) (−1)^r2^2rr!, where H_r(y) is the Hermite polynomial (see [2],[3]).

(14) L

1

r2(y²) = H2r+1(z) (−1)^r2^2r+1r!y. Also, we have:

(15) L^(α)_r (y) = (1 + α)_r

r! y⁻¹²⁻^a² exp y

2 M_r+¹

2+^a₂,^a₂(y) and

(16) L^(α)_r (y) = (−1)^r

r! y⁻¹²⁻^a² exp y

2 W_r+1

2+^a₂,^a₂(y),

where Ml,µ(y) and Wl,µ(y) are the Whittaker functions of first and second kind introduced by Whittaker [1] and defined by

(17) M_l,µ(y) = y^µ+¹²exp − y

2 F 1

2 + µ − l, 2µ + 1; y

,

(18) Wl,µ(y) = y^µ+¹² exp −y

2 U 1

2 + µ − l, 2µ + 1; y

.

In 1969, the multivariable hypergeometric function firstly introduced and studied by Srivastava and Daoust, generally known as Srivastava- Daust function which is given underneath as (see [5]):

F_l:q^p:m¹^;...;m^r

1;...;qr







(a_j : α¹_j, ...α^(r)_j )_1,p: (c¹_j, r¹_j)_1,q₁; ...; (c^(r)_j , r_j^(r))_1,q_r; (b_j : β_j¹, ...β^(r)_j )_1,l : (d¹_j, δ_j¹)_1,m₁; ...; (d^(r)_j , δ^(r)_j )_1,m_r;

x₁, x₂, ...x_r







(5)

(19)

=

∞

X

n1,n2,...nr=0 p

Q

j=i

(aj)_n

1a¹_j+...+nra^(r)_j q1

Q

j=i

(c⁽¹⁾_j )_n

1r⁽¹⁾_j ...

qr

Q

j=i

(c^(r)_j )_n

rr^(r)_j l

Q

j=i

(b_j)

n1b¹_j+...+nrb^(r)_j m1

Q

j=i

(d⁽¹⁾_j )

n1δ_j⁽¹⁾...

mr

Q

j=i

(d^(r)_j )

nrδ_j^(r)

xⁿ₁¹ n1!...xⁿ_r^r

nr!,

the multiple series (19) converges absolutely under the precise conditions (see, [6]).

We are required the most useful result due to Prudnikov et al [14]

by means of which we have established our main results in the present investigation:

(20)

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µdt

= (b − a)⁻¹(1 + k₁)^−λ(1 + k₂)^−µ Γ(λ)Γ(µ) Γ(λ + µ)

provided that <(λ) > 0, <(µ) > 0, b 6= a and the constants u and v are such that none of the expression 1+k₁, 1+k₂, [(b−a)+k₁(t−a)+k₂(b−t)], (where a ≤ b) is zero.

2. Main results

We establish two fascinating integral formulas, which are explicitly written in terms of Srivastava and Daust functions, by introducing product of generalized Bessel function with Jacobi and Laguerre polynomials with proper arguments in the integrand of (20).

This section deals with two intriguing integrals involving the product of generalized Bessel function with Jacobi and Laguerre polynomials, which are expressed in terms of Srivastava and Daust functions.

Theorem 2.1. Let λ and µ such that <(λ) > 0, <(µ) > 0 and the constants u and v such that [(b − a) + k₁(t − a) + k₂(b − t)], (where a ≤ b) is non-zero, then the underlying integral ( in terms of Srivastava and Daoust functions) holds:

Z _b

a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k2(b − t)λ+µw_σ,c^b

(6)

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×P_r^(α,β) 1 − 8(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)2

! dt

= 2^2σΓ(λ + σ)Γ(µ + σ)

Γ(λ + µ + 2σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k2)^µ+σ(b − a)

×F_5:0;1^4:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (1 + α + β : 1, 2) :

(M (2; λ + µ + 2σ) : 2, 3), (σ + ^1+b₂ : 1, 1), (1 + α + β : 1, 1) : (1 + α : 1, 1), −; −;

(1 : 1, 1), −; (1+α, 1);

−c

(1 + k₁)²(1 + k₂)², c

(1 + k₁)³(1 + k₂)³



,

where w_σ,c^b (t) and Pr^(α,β)(t) are well-known generalized Bessel function and Jacobi polynomials defined in (2) and (4), respectively, M (r; ζ) abbreviates of r parameters ^ζ_r,^ζ+1_r , . . . ,^ζ+r−1_r , r ≥ 1

Proof. For making the derivation of the Theorem (2.1), we symbolize the l.h.s. of Theorem (2.1) by I^∗, writing w^b_σ,c(t) and Pr^(α,β)(t) in their summation formula in the integrand with the help of equation (2) and (4) respectively, to obtain

I^∗ = Z _b

a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k2(b − t)λ+µ

×

∞

X

r=0

(−1)^r(c)^r[4(t − a)(b − t)]^σ+2r

r!Γ(σ + r + (^1+b₂ ))(b − a) + k₁(t − a) + k₂(b − t)2σ+4r

(21)

×

r

X

l=0

(1 + α)r(1 + α + β)r+l[−4(t − a)(b − t)]^l

l!(r−l)!(1+α)_r(1+α+β)_r+l(b−a)+k₁(t−a)+k₂(b− t)2ldt.

The above equation (21) directly follows from the fact given underneath:

(22)

∞

X

r=0 r

X

l=0

D(r, l) =

∞

X

r=0

∞

X

l=0

D(r, r + l)

(7)

and interchanging the order of integration and summation(which is authentic in the given condition), we have

I^∗ = 2^2σ

∞

X

r,l=0

(−c)^r(c)^l(1 + α)r+l(1 + α + β)r+2l 2^4r+6l

(r + l)! r! l! Γ(σ + r + l + (^1+b₂ ))(1 + α)l(1 + α + β)r+l

× Z b

a

(t − a)λ+σ+2r+3l−1(b − t)µ+σ+2r+3l−1

(b − a) + k₁(t − a) + k2(b − t)λ+µ+2σ+4r+6ldt .

Employing the result (20) and after some simplification, we obtain

I^∗ = 2^2σ Γ(λ + σ)Γ(µ + σ)

(b − a)Γ(λ + µ + 2σ)Γ σ +^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ+σ

(23)

×

∞

X

r,l=0

−c (1+k₁)²(1+k₂)²

r c

(1+k₁)³(1+k₂)³

l

(λ + σ)_(2r+3l)(µ + σ)_(2r+3l)(1 + α)r+l(1 + α + β)r+2l

(1)r+lr! l! (1 + α)l(1 + α + β)r+l(σ +^1+b₂ )r+l

λ+µ+2σ 2

2r+3l

λ+µ+2σ+1 2

2r+3l

.

Finally, after some simplification, summing up the above series with the aid of (19), we can easily obtain the desired result (2.1).

Theorem 2.2. Let λ and µ such that <(λ) > 0, <(µ) > 0 and the constants u and v such that [(b − a) + k₁(t − a) + k₂(b − t)], (where a ≤ b) is non-zero, then the underlying integral ( in terms of Srivastava and Daoust functions) holds:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k2(b − t)λ+µw_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×L^(α)_r 4(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= 2^2σΓ(λ + σ)Γ(µ + σ)

×F_4:0;1^3:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (1 + α : 1, 1) : (M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1) :

−; −;

(1 : 1, 1), −; (1 + α, 1);

−c

(1 + k₁)²(1 + k₂)², c

(1 + k₁)³(1 + k₂)³



,

(8)

where L^(α)_r (t) is the generalized Laguerre polynomial defined in (11).

Proof. For making the derivation of the Theorem (2.2), we symbolize the l.h.s. of Theorem (2.2) by I^∗∗, writing w^b_σ,c(t) and L^(α)r (t) in their summation formula in the integrand with the help of equation (2) and (11) respectively, to obtain

I^∗∗= Z b

a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µ

×

∞

X

r=0

(−1)^r(c)^r[4(t − a)(b − t)]^σ+2r

r!Γ(σ + r + (^1+b₂ ))(b − a) + k₁(t − a) + k2(b − t)2σ+4r

(24)

×

r

X

l=0

(−1)^l(1 + α)r(−4)^l(t − a)^l(b − t)^l

l!(r − l)!(1 + α)_l(b − a) + k₁(t − a) + k₂(b − t)2ldt.

Now applying the result (22) in equation (24) and then by interchanging the order of integration and summation(which is authentic in the given interval), we have

I^∗∗= 2^2σ

∞

X

r,l=0

(−c)^r(c)^l(1 + α)r+l 2^4r+6l

(r + l)! r! l! Γ(σ + r + l + (^1+b₂ ))(1 + α)l

× Z b

a

(t − a)λ+σ+2r+3l−1(b − t)µ+σ+2r+3l−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µ+2σ+4r+6ldt .

Employing the result (20) and after some simplification, we obtain

I^∗∗= 2^2σ Γ(λ + σ)Γ(µ + σ)

(b − a)Γ(λ + µ + 2σ)Γ σ +^1+b₂ (1 + k₁)^λ+σ(1 + k2)^µ+σ (25)

×

∞

X

r,l=0

−c (1+k1)²(1+k2)²

r c

(1+k1)³(1+k2)³

l

(λ + σ)_(2r+3l)(µ + σ)_(2r+3l)(1+α)_r+l (1)r+l r! l! (1 + α)l(σ +^1+b₂ )r+l

λ+µ+2σ 2

2r+3l

λ+µ+2σ+1 2

2r+3l

.

Finally, after some simplification, summing up the above series with the aid of (19), we can easily obtain the desired result (2.2).

Theorem 2.3. Let λ and µ such that <(λ) > 0, <(µ) > 0 and the constants u and v such that [(b − a) + k₁(t − a) + k₂(b − t)], (where a ≤ b)

(9)

is non-zero, then the underlying integral ( in terms of Srivastava and Daoust functions) holds:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µw_σ,c^b

× (t − a)

(b − a) + k₁(t − a) + k2(b − t)

!

×P_r^(α,β) 1 − (t − a)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= 2^−2σΓ(λ + σ)Γ(µ)

Γ(λ + µ + σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ(b − a)

×F_5:0;1^3:0;0





(λ + σ : 2, 3), (1 + α + β : 1, 2) :

(λ + µ + σ2 : 2, 3), (σ + ^1+b₂ : 1, 1), (1 + α + β : 1, 1) :

−; −; −;

(1 : 1, 1), −; (1 + α, 1);

−c

4(1 + k1)², c 8(1 + k1)³



, where w_σ,c^b (t) and P_r^(α,β)(t) are well-known generalized Bessel function and Jacobi polynomial defined in (2) and (4), respectively.

3. Particular cases

By making use of the above three Theorems (2.1), (2.2) and (2.3) through this segment, we establish some fascinating integral formulas associated with the product of Bessel function with ultraspherical polynomials, Gegenbauer polynomials, Tchebicheff polynomials and Legendre polynomials as a particular cases of our main results.

Corollary 3.1. On taking α = β in Theorem (2.1) and with the help of (6) then the underlying integral (which is guaranteed under the given conditions) holds true:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k2(b − t)λ+µw_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

(10)

×P_r^(α,α) 1 − 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= 2^2σΓ(λ + σ)Γ(µ + σ)

×F_5:0;1^4:0;0

"

(λ + σ : 2, 3), (µ + σ : 2, 3), (1 + 2α : 1, 2) :

(M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1), (1 + 2α : 1, 1) :

(26)

(1 + α : 1, 1), −; −;

(1 : 1, 1), −; (1 + α, 1);

−c

(1 + k1)²(1 + k2)², c (1 + k1)³(1 + k2)³



 ,

where Pr^(α,α) is ultraspherical polynomial (see [2, 3]).

Corollary 3.2. On replacing α = β = δ − ¹₂ in Theorem (2.1) and with the help of (7) then the underlying integral (which is guaranteed under the given conditions) holds true:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µw_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×C_r^δ 1 − 8(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)2

! dt

= 2^2σ(2δ)rΓ(λ + σ)Γ(µ + σ)

Γ(λ + µ + 2σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k2)^µ+σ(b − a)(δ + ¹₂)r

×F_5:0;1^4:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (2δ : 1, 2) :

(M (2; λ + µ + 2σ) : 2, 3), (σ + ^1+b₂ : 1, 1), (2δ : 1, 1) :

(27)

(¹₂+ δ : 1, 1), −; −;

(1 : 1, 1), −; (¹₂+ δ, 1);

−c

(1 + k1)²(1 + k2)², c (1 + k1)³(1 + k2)³



 ,

where C_r^δ(t) is the Gegenbauer polynomial (see [2, 3]).

Corollary 3.3. On substituting α = β = ¹₂ in Theorem (2.1) and with the help of (9) then the underlying integral (which is guaranteed under the given conditions) holds true:

Z _b

a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k2(b − t)λ+µw_σ,c^b

(11)

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×U_r 1 − 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= (n + 1)! 2^2σΓ(λ + σ)Γ(µ + σ)

3 2

rΓ(λ + µ + 2σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ+σ(b − a)

×F_5:0;1^4:0;0

"

(λ + σ : 2, 3), (µ + σ : 2, 3), (2 : 1, 2) :

(M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1), (2 : 1, 1) :

(28)

(³₂ : 1, 1), −; −;

(1 : 1, 1), −; (³₂, 1);

−c

(1 + k1)²(1 + k2)², c (1 + k1)³(1 + k2)³



 ,

where Ur(t) is Tchebycheff polynomial (see [2, 3]).

Corollary 3.4. On letting α = β = 0 in Theorem (2.1) and with the help of (10) then the underlying integral (which is guaranteed under the given conditions) holds true:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µw_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×P_r 1 − 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= 2^2σΓ(λ + σ)Γ(µ + σ)

×F_5:0;1^4:0;0

"

(λ + σ : 2, 3), (µ + σ : 2, 3), (1 : 1, 2) :

(M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1), (1 : 1, 1) :

(29)

(1 : 1, 1), −; −;

(1 : 1, 1), −; (1, 1);

−c

(1 + k1)²(1 + k2)², c (1 + k1)³(1 + k2)³



 ,

where P_r(t) is the Legendre polynomial (see[2, 3]).

(12)

Corollary 3.5. On applying the relation (15) in Theorem(2.2), then the underlying integral (which is guaranteed under the given conditions) holds:

Z b a

(t − a)^λ−^α²⁻³²(b − t)^µ−^α²⁻³²

(b − a) + k₁(t − a) + k₂(b − t)λ+µ−α−1w_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×e

4(t−a)(b−t)

[(b−a)+u(t−a)+v(b−t)]2M_r+1 2+^α₂,^α₂

4(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)2

! dt

= 2^2σ+α+1 r! Γ(λ + σ)Γ(µ + σ)

(1 + α)_rΓ(λ + µ + 2σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ+σ(b − a)

×F_4:0;1^3:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (1 + α : 1, 1) : (M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1) : (30)

−; −;

(1 : 1, 1), −; (1 + α, 1);

−c

(1 + k₁)²(1 + k₂)², c

(1 + k₁)³(1 + k₂)³



, where M_l,µ(y) denotes the Whittaker function of first kind (see [1]).

Corollary 3.6. On applying the relation (16) in Theorem(2.2), then the underlying integral (which is guaranteed under the given conditions) holds:

Z b a

(t − a)^λ−^α²⁻³²(b − t)^µ−^α²⁻³²

(b − a) + k₁(t − a) + k₂(b − t)λ+µ−α−1w_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×e

4(t−a)(b−t)

[(b−a)+u(t−a)+v(b−t)]2W_r+1 2+^α₂,^α₂

4(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= 2^2σ+α+1 r! Γ(λ + σ)Γ(µ + σ)

(−1)^r Γ(λ + µ + 2σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ+σ(b − a)

×F_4:0;1^3:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (1 + α : 1, 1) : (M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1) :

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−; −;

(1 : 1, 1), −; (1 + α, 1);

−c

(1 + k₁)²(1 + k₂)², c

(1 + k₁)³(1 + k₂)³



, where Wl,µ(y) denotes the Whittaker function of second kind (see [1]).

Corollary 3.7. Substituting α = −¹₂ in Theorem (2.2) and employing the relation(13), then the underlying integral (which is guaranteed under the given conditions) holds:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µw_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×H_2r 2p(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)

! dt

= 2^2σ+2r(−1)^r r! Γ(λ + σ)Γ(µ + σ)

Γ(λ + µ + 2σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ+σ(b − a)

×F_4:0;1^3:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (¹₂ : 1, 1) : (M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1) :

−; −;

(1 : 1, 1), −; (¹₂, 1);

−c

(1 + k1)²(1 + k2)², c

(1 + k1)³(1 + k2)³



, where Hr(y) is the Hermite polynomial (see [2, 3]).

Corollary 3.8. Substituting α = ¹₂ in Theorem (2.2) and employing the relation(14), then the underlying integral (which is guaranteed under the given conditions) holds:

Z _b

a

(t − a)^λ−³²(b − t)^µ−³²

(b − a) + k₁(t − a) + k2(b − t)λ+µ−1w_σ,c^b

× 8(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)2

!

×H_2r 2p(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)

! dt

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= 2^2σ+2r+2(−1)^r r! Γ(λ + σ)Γ(µ + σ)

×F_4:0;1^3:0;0





(λ + σ : 2, 3), (µ + σ : 2, 3), (³₂ : 1, 1) : (M (2; λ + µ + 2σ) : 2, 3), (σ +^1+b₂ : 1, 1) :

−; −;

(1 : 1, 1), −; (³₂, 1);

−c

(1 + k₁)²(1 + k₂)², c

(1 + k₁)³(1 + k₂)³



, where H2r+1(y) is again the Hermite polynomial (see [2, 3]).

Corollary 3.9. On replacing α = β = δ − ¹₂ in Theorem (2.3) and with the help of (7) then the underlying integral (which is guaranteed under the given conditions) holds true:

Z b a

(t − a)^λ−1(b − t)^µ−1

(b − a) + k₁(t − a) + k₂(b − t)λ+µw_σ,c^b

× (t − a)

(b − a) + k₁(t − a) + k₂(b − t)

!

×C_r^δ 1 − (t − a)

(b − a) + k₁(t − a) + k₂(b − t)

! dt

= 2^−2σΓ(λ + σ)Γ(µ)

Γ(λ + µ + σ)Γ σ + ^1+b₂ (1 + k₁)^λ+σ(1 + k₂)^µ(b − a)

×F_5:0;1^3:0;0





(λ + σ : 2, 3), (2δ : 1, 2), (δ +¹₂ : 1, 1),

(λ + µ + 2σ : 2, 3), (σ + ^1+b₂ : 1, 1), (2δ : 1, 1) :

−; −; −;

(1 : 1, 1), −; (δ +¹₂, 1);

−c

4(1 + k₁)², c 8(1 + k₁)³



, where C_r^δ(t) is Gegenbauer polynomial (see [2, 3]).

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4. Concluding Remaks

• This section deals with the disparity of the results obtained in the foregoing sections. The Meijer G-function [18] originated in 1936 is a general function encompassing the particular cases of the most of the known special functions. Further, we know that Bessel function of the first kind can be easily expressed in terms of Meijer G-function as follows:

Jσ(y) = G^1,0_0,2

−

σ 2, −^σ₂

y² 4

.

Similarly, the generalized Bessel function can also be written in terms of Meijer G-function as given below

w_σ,c^b (y) = y 2

σ

G^0,2_0,0

−

0, 1 − σ − (^1+b₂ )

−cy² 4

.

Therefore, the results depicted in the foregoing sections can easily be transformed in to Meijer G-function.

• With the aid of Theorems 2.1 and 2.2 we can also derived the fascinating results in which integrals involving the product of trigono- metric functions(sin and cos), hyperbolic functions (sinh and cosh) with orthogonal polynomials as shown here under.

• On taking α = β = 1, σ = ^b₂ and b = 1 and substituting c² in place of c by in Theorem (2.1) and using the Table 1, then the underlying integral (which is guaranteed under the given conditions) holds:

Z b a

(t − a)^λ−³²(b − t)^µ−³²

(b − a) + k₁(t − a) + k₂(b − t)λ+µ−1

× sin 8c(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

!

×P_r^(α,β) 1 − 8(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)2

! dt

= 8Γ(λ +¹₂)Γ(µ +¹₂)

√πΓ(λ + µ + 1)(1 + k1)^λ+¹²(1 + k2)^µ+¹²(b − a)

×F_5:0;1^4:0;0

"

(λ +¹₂ : 2, 3), (µ +¹₂ : 2, 3), (3 : 1, 2) :

(M (2; λ + µ + 1) : 2, 3), (³₂ : 1, 1), (3 : 1, 1) :

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(2 : 1, 1), −; −;

(1 : 1, 1), −; (2, 1);

−c²

(1 + k1)²(1 + k2)², c²

(1 + k1)³(1 + k2)³

# .

• On taking α = 1, σ = ^b₂ and b = 1 and substituting c² in place of c in Theorem (2.2) and using the Table 1, then the underlying integral (which is guaranteed under the given conditions) holds:

Z b a

(t − a)^λ−³²(b − t)^µ−³²

(b − a) + k₁(t − a) + k2(b − t)λ+µ−1

× sin 8c(t − a)(b − t)

(b − a) + k₁(t − a) + k₂(b − t)2

!

×L^(α)_r 1 − 4(t − a)(b − t)

(b − a) + k₁(t − a) + k2(b − t)2

! dt

= 8Γ(λ +¹₂)Γ(µ +¹₂)

√πΓ(λ + µ + 1)(1 + k1)^λ+¹²(1 + k1)^µ+¹²(b − a)

×F_5:0;1^4:0;0

"

(λ +¹₂ : 2, 3), (µ +¹₂ : 2, 3), (3 : 1, 2) :

(M (2; λ + µ + 1) : 2, 3), (³₂ : 1, 1), (3 : 1, 1) :

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(2 : 1, 1), −; −;

(1 : 1, 1), −; (2, 1);

−c²

(1 + k1)²(1 + k2)², c² (1 + k1)³(1 + k2)³



 .

References

[1] E. T. Whittaker and G. N. Watson, A course of modern analysis , 4th edition, Cambridge Univ. Press, Cambridge, London, Now york, 1927.

[2] E. D. Rainville, Special Functions. Macmillan Company, New York, (1960);

Reprinted by Chelsea Publ.co. Bronx, New York, 1971.

[3] H. M. Srivastava, and H. L. Manocha, A Treatise on Generating Functions.

Chichester, New York; Ellis Horwood Limited, John Wiley and Sons, New York, 1984.

[4] H. M. Srivastava and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1985.

[5] H. M. Srivastava and M. C. Daust, Certain generalized Neumann expansions associated with the Kamp´e de F´eriet function, Nederl. Akad. Wetensh. Proc.

Ser. A 72 = Indag Math. 31 (1969), 449-457 .

[6] H. M. Srivastava and M. C. Daust, A note on the convergence of Kamp´e de F´eriet’s double hypergeometric series , Math. Nachr. 53 (1972), 151-159.

[7] J. Choi and P. Agarwal, Certain unified integrals associated with Bessel functions, Boundary value problems, 1 (2013), pp.95.

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[8] J. Choi and P. Agarwal, Certain unified integrals involving a product of Bessel function of first kind, Honam Mathematical Journal, 35(4), (2013), 667-677.

[9] J. Choi, P. Agarwal, S. Mathur, and S.D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bulletin of the Korean Mathematical Society 51(4) (2014), 995-1003 .

[10] M. Ghayasuddin, N. U. Khan, and S. W. Khan, Some finite integrals involving the product of Bessel function with Jacobi and Laguerre polynomials , Commu- nications of the Korean Mathematical Society, (2017), (Accepted).

[11] N. U. Khan, M. Ghayasuddin and T. Usman, On certain integral formulas involving the product of Bessel function and Jacobi polynomial, Tamkang Journal of Mathematics.,47(3) (2016), 151-153 .

[12] N. U. Khan, T. Usman and M. Ghayasuddin, A Unified double integral associated with Whittaker functions , Journal of Nonlinear Systems and Applications (2016) 21-24.

[13] N. U. Khan, T. Usman and M. Ghayasuddin, A new class of unified integrals formulas associated with whittaker functions , New Trends in Mathematical Sciences 4(1) (2016), 160-167.

[14] Prudnikov, A. P., Brychkov, Yu. A. and Marichev,O. I. Integral and Series V.3.

More Special Functions, New York-London: Gordon and Breach, 1992.

[15] P. Agarwal, S. Jain, S. Agarwal, and M. Nagpal, On a new class of integrals involving Bessel functions of the first kind, Communications in Numerical Analal- ysis (2014), 1-7 .

[16] S. Ali, On some new unified integrals, Adv. Comput. Math. Appl., 1 (2012), 151-153.

[17] Saiful R. Mondal, A. Swaminathan, Geometric Properties of Generalized Bessel Functions, Bulletin of the Malaysian Mathematical Sciences Society (2) 35(1) (2012), 179-194.

[18] V. Adamchik, The Evaluation of Integrals of Bessel Functions via G-Function Identities, Journal of Computational and Applied Mathematics 64 (1995) 283- 290.

Nabiullah Khan

Department of Applied Mathematics, Faculty of Engineering and Tech- nology,Aligarh Muslim University,

Aligarh-202002, India.

E-mail: [email protected] Raghib Nadeem

E-mail: [email protected]

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Talha Usman

E-mail:[email protected] Abdul Hakim Khan

E-mail:[email protected]