Vol 9, No 1 (2018)

(1)

Available online throug

ISSN 2229 – 5046

FRACTIONAL INTEGRALS

INVOLVING

ℵ

-FUNCTION AND THE GENERAL CLASS OF POLYNOMIALS

MONIKA JAIN

Department of Mathematics,

JECRC University, Jaipur-303905, Rajasthan, India.

(Received On: 11-12-17; Revised & Accepted On: 03-01-18)

ABSTRACT

The object of the present paper is to study a number of new and useful fractional integrals involving a product of three

general class of polynomials and the Aleph(ℵ)- function with xk (xh + dh)–w as general arguments in view of both the operators introduced by Saigo in 1978.

The fractional integrals established here are quite general in character and on specializing the parameters of Aleph(ℵ)- function and the arbitrary coefficients occurring in three general class of polynomials, a large number of fractional integrals involving various classes of orthogonal polynomials, generalized hypergeometric polynomials and elementary functions (or product of several such functions) can be obtained from them. Thus, our results provide interesting unifications and extensions of a number of known and new results. Some special cases have also been recorded.

2010 Mathematics Subject Classifications: 26A33, 33E20, 33C20, 33C45.

Key words: Aleph function, Fractional Calculus, Mellin-Barnes type integrals, General class of Polynomials,

H-function, I-function.

1. INTRODUCTION AND PRELIMINARIES

The Aleph-function is a new generalization of the familiar H-function [11] and the I-function [15].

The Aleph-function is defined by means of a Mellin-Barnes type integral in the following manner [13, 15]:

(

)

(

)

(

)

(

)

( )

, ,..., , _,

1, 1,

,

, , ; _, _,..., _,

, , ;

1, 1,

1 2

[ ]

:

a_j A_j _j a_j A_j _{m n}

n n p

m n i s

p q_i _{i i}r _b _B _b _B _L

p q r

j j _m j j j _m _qi i i i i

z

s z

ds

τ

τ _τ _τ

π

 

  + −

 

  +





ℵ

= ℵ





=

Ω









∫

, (1)

where

z

≠

0

,

i

= −

1

and

(

) (

)

(

)

(

)

1 1

, , , ;

1 1 1

.

1 ( )

1 .

= =

= = + = +

Γ +

Γ − −

Ω

=

Γ − −

Γ

+

∏

∑ ∏

∏

m n

j j j j

j j

m n

p qi i ir r qi pi

i ji ji ji ji

i j m j n

b

B s

a

A s

s

b

B s

a

A s

τ

. (2)

The integration path

L

=

L

iγ∞

,

(

γ

∈ ℜ

)

extends from

γ − ∞

i

to

γ + ∞

i

, and is such that the poles of

(

1 a

_j

A s

_j

)

,

j

( )

1,

n

Γ −

−

=

(the symbol

( )

1,

n

is used for 1, 2,…,n) do not coincide with the poles of

(

b

j

B s

j

)

,

j

( )

1,

m

Γ

+

=

. The parameters

p q

i

,

i are non-negative integers satisfying the condition

0 ≤ ≤

n

p

i

, 1

≤ ≤

m

q

i

,

τ

i

>

0

for

i

=

1,

r

. The parameters

A B A B

j

,

j

,

ji

,

ji

>

0

and

a b a b

j

,

j

,

ji

,

ji

∈

C

. The empty product in (2) is interpreted as unity. The existence conditions for the defining integral (1.1) are given below:

Corresponding Author: Monika Jain

(2)

0,

arg( )

,

1, ;

2

l

z

l

r

π

ϕ

>

<

ϕ

=

(3)

0,

arg( )

2

l

z

l

π

ϕ

≥

<

ϕ

and

ℜ

{ } 1 0 ,

ζ

_l

+ <

(4)

Where

1 1 1 1

p q

n m l l

l j j l jl jl

j j j n j m

A

B

A

B

ϕ

τ

= = = + = +





=

+

−



_

+



_





∑

(5)

(

)

1 1 1 1

1 ,

1, .

2

q p

m n l l

l j j l jl jl l l

j j j m j n

b

a

b

a

p

q

l

r

ζ

τ

= = = + = +





=

−

+



_

−



_

+

−

=





∑ ∑

∑

(6)

Remark 1: For

τ

_i

=

1,

i

=

1,

r

in (1), we get the I-function due to V.P. Saxena [18], defined in the following manner:

( ) ( )

( )

, ,..., , ,

1, 1,

, , ,

, ; , ,1; , ,1; _, _,..., _,

, ,1;

1, 1,

1

2 [ ]

[ ]

:

a Aj j _n a Aj j _n _p m n

m n m n m n _i s

p q r_{i i} p q_{i i} r p q_{i i} r _{b B} _{b B} _L

j j _m j j _m p q_{i i} r

qi

i

I

z

s z ds

π

+ −

+





= ℵ

=ℵ

_

_

=

Ω





∫

, (7)

where the kernel _{p q}m n,_, _,1;_r

( )

i i

s

Ω

is given in (2). The existence conditions for the integral in (7) are the same as given in (3) - (6) with

τ

_i=1, i=1,r.

Remark 2: If we further set r = 1, then (7) reduces to the familiar H- function [3,10,12]:

[ ]

₍

(

,

₎

)

( )

, , , ,

, , ,1;1 , ,1;1 _, , ,1;1

1 :

2

p p

i i i i _q _q i i

a A

m n m n m n m n s

p q p q p q _{b B} p q

L

H

z

s z

ds

i

π

−





= ℵ

=ℵ

_

_

=

Ω





∫

, (8)

where the kernel _{p q}m n_,, _,1;1

( )

i i

s

Ω

can be obtained from (2).

Remark 3: Fractional integration of Aleph function is discussed by Saxena and Pogány [16]. A detailed account of

ℵ

- function is given in the papers by Saxena et.al [9, 16, 17].

The series representation of Aleph(ℵ)- function is given by

ℵ_𝑝

𝑖

′_,_𝑞

𝑖

′_,_𝑐

𝑖

′_;_𝑟′

𝑚′,𝑛′ _[_𝑢_{] =}_ℵ 𝑝_𝑖′,𝑞_𝑖′,𝑐_𝑖′;𝑟′ 𝑚′,𝑛′ _�𝑢_|(𝑎𝑗

′_,_𝐴

𝑗′)1,𝑛′,�𝑐𝑖′(𝑎𝑗𝑖′ ,𝐴𝑗𝑖′ )𝑛′+1,𝑝_𝑖′ ;𝑟′�

(𝑏𝑗′,𝐵𝑗′)1,𝑚′, �𝑐𝑖′(𝑏𝑗𝑖′ ,𝐵𝑗𝑖′ )𝑚′+1,𝑞_𝑖′ ;𝑟′�

�

= ∑ ∑ (−1)

𝑘′_∅′₍_𝜂

ℎ,𝑘′) 𝑢−𝜂ℎ,𝑘′

𝐵_ℎ′ 𝑘′_!

∞

𝑘′₌₀

𝑚′

ℎ=1 , (9)

where ∅′(𝜂_ℎ_,_𝑘′) = ∏ Γ(𝑏𝑗 ′_+𝐵

𝑗′𝜂ℎ,𝑘′)∏𝑛 Γ(1−𝑎′𝑗−𝐴𝑗′𝜂ℎ,𝑘′)

′

𝑗=1 𝑚′

𝑗=1

∑𝑟_𝑖=1′ 𝐶_𝑖′�∏𝑞𝑖 Γ(1−𝑏_𝑗𝑖′−𝐵_𝑗𝑖′𝜂_ℎ_,_𝑘′)∏_𝑗=1𝑛′ Γ(𝑎_𝑗𝑖′ +𝐴′_𝑗𝑖𝜂_ℎ_,_𝑘′)� ′

𝑗=𝑚′+1

, (10)

and 𝜂_ℎ_,_𝑘′ = 𝑏ℎ′+𝑘′

𝐵_ℎ′ , 𝑝𝑖′ <𝑞𝑖′ , |𝑢| < 1 .

FRACTIONAL INTEGRALS

Let α, β and η are complex numbers and let y ∈ R+ (0, ∞). Following [5, 14, 15] the fractional integral (Re (α) > 0) and derivative (Re (α) < 0) of the first kind of a function f(y) on R+ are defined respectively in the following forms

( )

α β

α,β.η

y

I

f

α

0,y

− −

=

Γ

(

)

y

_{α 1}

y t

F

2 1

0 −

−

∫

α β, η;α;1

t

y











+ −

−



f t dt ; Re

( )

α

>

0

(11)

n

d

n

dy

=

I

α n,β n,η n

f,

0,y

+

−

0 <

Re

( )

α +

n

≤

1 (

n

=

1, 2,...

)

(12)

(3)

The fractional integral (Re (α) > 0) and derivative (Re (α) < 0) of the second kind are given by

( )

α,β,η

1 J

f

y,

∞

=

Γ

α

α 1

(t

y)

y

∞

₋

−

∫

t

α β

F

2 1

− −

y

α β, η;α;1

t











+ −

−



f t dt, Re

( )

α

>

0

(13)

n

_n,

d

n

( 1)

_{n y,}

I

f ,

dy

α

+

β

−

η

= −

_∞

0 <

Re ( ) n

α

+ ≤

1 (n

=

1,2,...)

(14)

A GENERAL CLASS OF POLYNOMIALS

Srivastava [19, p., Eq. (1), 17] introduced the general class of polynomials

( N )

[N /M ]

₁

_{M K}

M

₁

1

1 _{1 1}

K

₁

S

_N

[x]

A

_{N ,K}

x

,

N

0,1, 2,...

1 K !

K

0

1 ₁

₁

1 1

−

=

∑

=

… (15)

where M1 is an arbitrary positive integer and the coefficients

A

_{N ,K}

(N , K

₁

0)

1 1

≥

are arbitrary constants, real or complex.

Here (λ)n denotes the Pochhamer symbol defined by

( )

(

_{( )}

)

(

) (

)

{

}

Γ λ +n

λ

_n

=

Γ λ

1,

if n

0 λ λ 1 ... λ n 1 ,

n 1,2,3,...







=

+

+ −

∀ ∈

By suitably specializing the coefficients

A

N , K

1

occurring in (15), the general class polynomials

M

1 S

[x]

N

1

can be

reduced to the classical orthogonal polynomials and the generalized hypergeometric polynomials.

(

)

F

α,β;γ;1

2 1

=

( ) (

)

(

γ

γ α

) (

γ α β

γ β

)

,

Γ

Γ − −

Γ − Γ −

Re (γ – α – β) > 0, Re (γ) > 0 (16)

(

)

F

α,β;γ;z

2 1

=

( ) ( )

( )

α

_n

β

_n

n z

γ n!

n

0 _n

∞

∑

=

… (17)

2. THE FOLLOWING TWO LEMMAS ARE PROVED TO EMPLOY IN THE SEQUEL

Lemma 1: If

Re (a)

0, k

,

b,c

>

∈Ω

d is a positive number and h = 1, 2, 3, … and w, p, q are complex numbers,

m1, m2 are arbitrary positive integers and the coefficients

A

_{n ,s}

(n ,s

0), A

_{n ,s}

(n ,s

0)

1 1

2 2

1 1

≥

2 2

≥

are arbitrary

constants, real or complex, then

m

a,b,c

_k

_h

_w

₁

_h

_p

₂

_h

_q

I

x (x

d )

S

_n

[yx

(x

d )

]S

_n

[Dx (x

d )

]

0,x

₁

₂

β

α

















−

+

[n /m ] [n /m ] ( n )

₁

₂

_{m s}

_s

( n )

_{m s}

_s

1 _{1 1}

₁

2 _{2 2}

₂

A

_{n ,s}

y

A

_{n ,s}

D

s !

1 1

s !

2 2

s

0 s

0 ₁

₂

1

2 −

−

=

∑

=

.

k

s

b

1

2 (k

s

1) (k

s

b c 1) x

1

2

1

2 hw hps

hqs

1

2 (k

s

b 1) (k

s

a c 1) d

1

2

1

2 α

β

α

β

α

β

α

β

α

β

+

−

Γ +

+

+ Γ +

+

− + +

+

(4)

.

k

s

1 k

s

h k

s

b c 1

1

2

1

2

1

2 w ps

qs ,

,...,

,

,...,

1

2 h

h

F

2h 1 2

hk

s

b 1

k

s

b h a k

s

c 1

1

2 _,...,

1

2 _,

1

2 _,...,

1 h

h

α

β

α

β

α

β

α

β

α

β

α

β







+

− + +

+

− +

+

− +

+ +

+

+ +

k

s

b c h

1

2 ;

_x

h

a k

s

₁

s

₂

c h

_d

h

;

h

α

β

α

β







+

− + +

−

+ +

+

+ +

(18)

Proof: To establish Lemma 1, we first operate the fractional integral operator by (11) for

m

₁

m

₂

p

q

k

h

w

h

f(t)

t (t

d )

S

_n

[y t (t

d )

]S

_n

[Dt (t

d )

]

1

2 β

α

−

=

+

(19)

and express Gauss function by (17) and

m

1

2 S

_n

,S

_n

1

2

by (15) with

h

w

(t

+

d )

−

in series form by the formula

h

(w)

t

h

w

hw

(t

d )

d

h

!

0 d

σ

















∞

−

+

=

∑

−

=

… (20)

Interchanging the order of summations and integration, this is permissible due to the absolute convergence involved in the process, evaluating the integral by the following result

x

_{m 1}

_kn

_m

_kn

₍

_{m) (}

_kn

_a)

(x

t)

t

dt

x

(

m

kn 1)

0 α

λ

α

λ

α

λ

α

λ

Γ +

+

+ −

+

+ + +

−

=

∫

_{Γ + + +}

₊

(21)

With a little simplification, we get

[n /m ] [n /m ] ( n )

₁

₂

_{m s}

_s

( n )

_{m s}

_s

1 _{1 1}

₁

2 _{2 2}

₂

A

_{n ,s}

y

A

_{n ,s}

D

s !

1 1

s !

2 2

s

0 s

0 ₁

₂

1

2 −

−

∑

=

.

k

s

b

1

2 (k

s

1)

x

₁

₂

hw hps

hqs

_(a

_s

_{k 1)}

1

2 ₁

₂

d

α

β

_α

_β

α

β

+

−

_{Γ +}

₊

+

Γ +

+

+ +

.

(k

s

1)

(w

ps

qs )

1

2 h

1

2 (a

s

k 1)

!

0 ₁

₂

_h

α

β

_σ

α

β

σ

_σ

+

∞

∑

+

+ +

=

h

x

a b , c

;

F

1 2 1 a k

s

h

1 ;

_d

h

1

2 σ

α

β

σ















_{ }

_







_{ }

_

+

−

₋

+ +

+

(22)

Then we get the desired result by applying Gauss theorem (17) and multiplication formula Rainville [13, cf. Theo.18] with a little simplification.

Lemma 2: If

Re (a)

0, k

,

b,c

>

∈Ω

d is a positive number and h = 1, 2, 3, … and w, p, q are complex numbers

m1, m2 are arbitrary positive integers and the coefficients

A

_{n ,s}

(n ,s

0), A

_{n ,s}

(n ,s

0)

1 1

2 2

1 1

≥

2 2

≥

are

arbitrary constants, real or complex, then

m

a,b,c

_k

_h

_w

₁

_h

_p

₂

_h

_q

J

_x,

x (x

d )

S

_n

[yx

(x

d )

]S

_n

[Dx (x

d )

]

1

2 β

α

















−

+

∞

[n /m ] [n /m ] ( n )

_{m s}

_s

( n )

_{m s}

_s

1

2

1 _{1 1}

₁

2 _{2 2}

₂

A

_{n ,s}

y

A

_{n ,s}

D

s !

1 1

s !

2 2

s

0 s

0 ₁

₂

1

2 −

−

=

∑

(5)

.

k

s

b

1

2 (b k

s

s ) (c k

s

s )

x

₁

₂

₁

₂

hw hps

₁

hqs

₂

( k

s

s ) (a

b c k

s

s )

1

2

1

2 d

α

β

_α

_β

_α

_β

α

β

α

β

+

−

_{Γ − −}

₋

_{Γ − −}

₋

+

_{Γ − −}

₋

_{Γ + + − −}

₋

.

k

s

1 k

s

h k

s

a b c 1

1

2

1

2

1

2 w ps

qs ,

,...,

,

,...,

1

2 h

h

F

2h 1 2

h

k

s

b 1

k

s

b h k

s

c 1

2 _,...,

1

2 _,

1

2 _,...,

1 h

h

α

β

α

β

α

β

α β

α

β

α

β







+

− − − +

+

+ +

− +

+

− +

+

− +

k

s

a b c h

1

2 ;

_x

h

k

s

₁

s

₂

c h

_d

h

;

h

α

β

α

β







+

− − − +

−

+

− +

… (23)

Proof: To prove Lemma 2, we take

m

1 p

2 q

k

h

w

h

f(t)

t (t

d )

S

_n

[y t (t

d )

]S

_n

[Dt (t

d )

]

1

2 β

α

−

=

+

in equation (13) and write series expansions for the Gauss function by (17) and

m

1

2 S

_n

,S

_n

1

2

by (15) with

h

w

(t

+

d )

−

,

then interchanging the order of integration and summations which is permissible due to the absolute convergence involved in the process, solve the integral by the following result

(

m) (

kn)

m kn

kn

m 1

(t

x)

t

dt

x

,

x

(

m

kn)

α

β γ

γ α β

γ β

α

α β

γ

∞

_{+ −}

_{− − − +}

_{− +}

_{Γ +}

_{Γ − −}

−

=

∫

_{Γ + + − −}

(24)

and using the relation

n

( 1)

(a) n

(1 a)n

−

=

−

₋

, (25)

We get

[n /m ] [n /m ] ( n )

_{m s}

_s

( n )

_{m s}

_s

1

2

1 _{1 1}

₁

2 _{2 2}

₂

A

_{n ,s}

y

A

_{n ,s}

D

s !

1 1

s !

2 2

s

0 s

0 ₁

₂

1

2 −

−

∑

=

.

(b k

s

s )

k

s

b

hw hps

hqs

₁

₂

1

2

1

2 x

d

(a

b k

s

s )

1

2 α

β

α

β

α

β

Γ − −

−

+

−

Γ + − −

−

.

(1 a b k

s

s )

(w

s

qs )

1 2 h

1

2 !

(1 b k

s

s )

0 ₁

_{2 h}

α

β

_σ

σ

α

β

σ

_σ

− − + +

+

∞

∑

− + +

+

=

h

a b,

c

;

_x

F

1 .

2 1

_{a b k}

_s

_h

_;

_d

h

1

2 σ

α β

σ



 





 









_{ }

_





+ −

−

+ − −

−

(26)

Now using Gauss theorem (4) and multiplication formula [13] and with a little simplification, we obtain the desired result.

3. THE FRACTIONAL INTEGRAL FORMULAE

If

m

_p

₂

_q

k

h

w

₁

h

f(t)

t (t

d )

S

_n

[y t (t

d )

]S

_n

[Dt

(t

d )

]

1

2 β

α

−

=

+

Sn3

m3_�_Etγ_(th_{+ d}h₎−r_�_._ℵ pi′,qi′,ci′;r′

m′,n′ _�_ztλ_(th_{+ d}h₎−n_�(aj

′_{, A}

j

′₎

1,n′,�ci′(a′ji, A′ji)n′+1,p_i′ ;r′�

(bj′, Bj′)1,m′, �ci′(bji′, Bji′)m′+1,qi′ ;r′�

(6)

then, we get

'

[n /m ]

[n /m ] [n /m ]

a,b,c

1

2

3

3 I

[f(x)]

(s ,s ,s , h,k )

0,x

_{s 0}

_s

₀

_s

₀

_{h 1 k 0}

1 2 3

1

2

3 m

ϕ

∞

=

∑

=

.

R b

x

(R 1) (R

c b 1)

T

d

(R

b 1) (R

c a 1)

− Γ + Γ + − +

Γ

− + Γ

+ + +

' h,k .

R 1

R h R c b 1

R c b h

w ps

qs

rs

n

,

,...,

,

,...,

;

_h

1

2

3 h

h

x

F

2h 1 2h R b 1

_,...,

R b h

_,

R c a 1

_,...,

R c a h

_;

_d

h

η

 ₋ 

 

 

+

+ − +

+

−

+

− +

+ + +

(27)

where

'

( n )

_{m s}

A

_{n ,s}

( n )

_{m s}

B

_{n ,s}

( n )

_{m s}

C

_{n ,s}

1 _{1 1}

_{1 1}

2 _{2 2}

_{2 2}

3 _{3 3}

_{3 3}

(s ,s ,s , h,k )

1 2 3

_{s !}

_{s ! s !}

1

2

3 ϕ

=

−

' h,k

'

' _'

' '

s

k

₁

₂

₃

( 1)

(

) y

D

E

z

h,k

,

k ! B

_h

η

ϕ η

−

×

'

,

R

k

s

1

2

3 η

h k

α

β

γ

−

λ

=

+

and

T

=

hw

+

hps

₁

+

hqs

₂

+

hrs

₃

−

hn

η

_{h k}_, ' .

The result in (27) is valid for Re (a) > 0, (k + αs1 + βs2 + γs3 + λfi/Fi) ∈Ωb,c, i = 1, …, M.

'

T

_'

|arg z |

, T

0

2 π

<

>

, d is a positive number and w, α, p, β, q, γ, r, λ, n are complex numbers, h = 1, 2, …, m1, m2, m3 are arbitrary positive integers and the coefficients

A

_{n ,s}

(n ,s

_{1 1}

0),

1 1

≥

B

n ,s

2 2

(n ,s (n ,s

2 2

≥

0),

C

n ,s

3 3

(n ,s

3 3

≥

0)

are arbitrary constants, real or complex.

'

[n /m ]

[n /m ] [n /m ]

_M

a,b,c

1

2

3

3 J

_x,

[f(x)]

(s ,s ,s , h,k )

1 2 3

s

0 s

0 G 0 g 1

1

2

3 ϕ

∞

=

∑

∞

₌

.

R b

x

(c R) (b R)

T

d

(a

b c R) ( R)

− Γ − Γ −

Γ + + −

Γ −

'

-.

R 1

R h R a b c 1

R a b c h

w ps qs

rs n

,

,...,

,

,...,

;

_x

h

1

2

3 h,k

h

F

2h 1 2h R c 1

_,...,

R c h

_,

R b 1

_,...,

R b h

_;

_d

h

η





















+

− − − +

+ +

+

−

+

− +

(28)

where d is a positive number and w, α, p, β, q, γ, r, λ, n are complex numbers, h = 1, 2, …, Re (a) > 0, (k + αs1 + βs2 +

γs3 + λfi/Fi) ∈Ωb, c, i = 1, …, M,

| arg z|

1 '

T ,

2 π

<

_T

'

_{0, m , m , m}

1

2

3 >

are arbitrary positive integers and the

coefficients

A

_{n ,s}

(n ,s

0),

1 1

≥

B

n ,s

2 2

(n ,s

2 2

≥

0),

C

n ,s

3 3

(n ,s

3 3

≥

0)

are arbitrary constants, real or

complex.

(7)

SPECIAL CASES

(I) If we take λ = 0 = n, the results in (27) and (28) reduces to the following formula

m

a,b,c

_k

_h

_w

₁

_h

_p

I

x (x

d )

S

_n

[yx (x

d )

]

0,x

₁

α





+

−

+

−

.

m

_h

_q

_h

_r

3

2 S

_n

[Dx (x

d )

] S

_n

[e x (x

d )

]

2

3 β

γ





−

+

[n /m ]

[n /m ] [n /m ]

( n )

k b

₁

₂

₃

₁

m s

₂

m s

₃

m s

x

_{1 1}

_{2 2}

_{3 3}

hw

_s

₀

_s

₀

_b

₀

s !

d

₁

₂

₃

₁

₂

₃

−

=

∑

=

.

s

₂

s

1

3 s

s

_x

3

1

2 A

_{n ,s}

B

_{n ,s}

C

_{n ,s}

y

D

E

hp

hq

hr

1 1

2 2

3 3

_d

β

γ

α























_



_



_{ }

_{ }

_



_{ }

_

.

(k

s

1) (k

s

c b 1)

1

2

3

1

2

3 (k

s

b 1) (k

s

c a 1)

1

2

3

1

2

3 α

β

γ

α

β

γ

α

β

γ

α

β

γ

Γ +

+

+ Γ +

+

+ − +

Γ +

+

− + Γ +

+

+ + +

.

k

s

1 k

s

h

1

2

3

1

2

3 w ps

qs

rs ,

,...,

,

1

2

3 h

h

F

2h 1 2h k

s

b 1

k

s

b h

1

2

3 _{, ...,}

1

2

3 _,

h

α

β

γ

α

β

γ

α

β

γ

α

β

γ







+

− +

+

− +

k

s

b c 1

k

s

b c h

1

2

3 ,...,

1

2

3 ;

_x

h

k

s

₁

s

₂

s

₃

c a 1

k

s

₁

s

₂

s

₃

c a h

_d

h

,...,

;

h

α

β

γ

α

β

γ

α

β

γ

α

β

γ







+

− + +

+

− + +

−

+

+ + +

+

+ + +

, (29)

and

m

a,b,c

_k

_h

_w

₁

_h

_p

J

_x,

x (x

d )

S

_n

[yx (x

d )

]

1 α





+

−

+

−

∞

.

m

_h

_q

_h

_r

3

2 S

_n

[Dx (x

d )

] S

_n

[E x (x

d )

]

2

3 β

γ





−

+

[n /m ]

[n /m ] [n /m ]

( n )

k b

₁

₂

₃

₁

m s

₂

m s

₃

m s

x

_{1 1}

_{2 2}

_{3 3}

hw

_s

₀

_s

₀

_b

₀

s !

d

₁

₂

₃

₁

₂

₃

−

=

∑

=

.

s

₂

s

1

3 s

s

_x

3

1

2 A

_{n ,s}

B

_{n ,s}

C

_{n ,s}

y

D

E

hp

hq

hr

1 1

2 2

3 3

_d

β

γ

α























_



_



_{ }

_{ }

_



_{ }

_

.

(c k

s

s ) (b k

s

s )

1

2

3

1

2

3 (a

b c k

s

s ) ( k

s

s )

1

2

3

1

2

3 α

β

γ

α

β

γ

α

β

γ

α

β

γ

Γ − −

−

Γ − −

−

Γ + + − −

−

Γ − −

−

k

s

1 k

s

h

1

2

3

1

2

3 w ps

qs

rs ,

,...,

,

1

2

3 h

h

F

2h 1 2h k

s

c 1

k

s

c h

1

2

3 _{, ...,}

1

2

3 _,

h

α

β

γ

α

β

γ

α

β

γ

α

β

γ







+

− +

+

− +

k

s

b a c 1

k

s

a b

c h

-1

2

3 ,...,

1

2

3 ;

_x

h

k

s

₁

s

₂

s

₃

b 1

k

s

₁

s

₂

s

₃

b h

_d

h

,...,

;

h

α β

γ

α β

γ

α β

γ

α β

γ







+

− − − +

+

− − +

−

(8)

(II) Letting w → 0, n1→ 0 and n2→ 0, Lemma 1 and Lemma 2 are reduced to the result of Saigo and Raina [11] and for h = y = D = E = 1, Lemma 1 and 2 with results (29) and (30) are reduced to the results of Banerji and Choudhary [1].

(III) For n1→ 0, n2→ 0, Lemma 1 and Lemma 2 along with the results (27) and (28) and suitably specializing the arbitrary coefficients of general class of polynomials are reduced to the known results obtained by Chaurasia and Gupta [6].

(IV) For

τ

_i

=

1,

i

=

1,

r

and further set r = 1, then the results (27) and (28) reduced to the results obtained by Chaurasia and Jain [2].

It may be of interest to remark that the formulae derived in this chapter may be useful in obtaining formulae for various classical orthogonal polynomials.

REFERENCES

1. Banerji, P. K. and Choudhary, Sudipto, Fractional integral formulae involving general class of polynomials, Proc. Nat. Acad. Sci. (India), 66A III (1996), 271-277.

2. Chaurasia and Jain. Investigating in to General Polynomials, in Thesis, University of Rajasthan, Jaipur, India,2008

3. Chaurasia, V. B. L. and Dubey, R. S., Analytical Solution for the Differential Equation Containing Generalized Fractional Derivative Operators and Mittag-Leffler-Type Function, Inte. Scho. Res. Net., ISRN App. Math., 2011, Article ID 682381, 9 pages doi:10.5402/2011/682381.

4. Chaurasia, V. B. L. and Dubey, R. S., Images of certain special functions pertaining to multiple Erd́elyi-Kober operator of Weyl type, scientia Series A: Mathematical Sciences, 20 (2009), 37–43

5. Chaurasia, V. B. L. and Dubey, R. S., The n- Dimensional Generalized Weyl Fractional Calculus Containing to n- Dimensional H-Transforms, Gen. Math. Notes, 6(1) (2011), 61-72.

6. Chaurasia, V. B. L. and Gupta, Neeti, General fractional integral operators, general class of polynomials and Fox’s H-function, Soochow J. Math. 25(4) (1999), 333-339.

7. Chaurasia, V.B.L. and Gulshan Chand, Certain integral properties of Aleph-function, (Communicated for publication).

8. Chaurasia, V.B.L. and Shekhawat, A.S., Some integral properties of a general class of polynomials associated with Feynman integrals, Bull. Malays. Math. Sci. Soc., (2) (28) (2) (2005), 183-189.

9. Dubey, R. S., An application of the aleph (ℵ)–function for detecting glucose supply in human blood, Inte. J. of Mod. Mathe. Sci., 14(3), 221-226.

10. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Tables of Integral Transforms, Vol. 2, McGraw-Hill, New York-London (1954).

11. Fox, C. The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429. 12. Gupta, K.C., Gupta, K. and Gupta, A., Generalized fractional integration of the product of two H-function, J.

Raj. Acad. Sci., 9(3) (2010):203-212.

13. Rainville, E. D.,Special Functions, Chelsea Publishing Co., Bronx, New York, 1971.

14. Saigo, M. and Raina R. K., Fractional Calculus operators associated with a general class of polynomials, Fukuoka Univ. Sci. Reports, 18 : 1 (1988), 15-22.

15. Saigo, M., A remark on integral operators involving the Gauss Hypergeometric Functions, Math. Rep. College General Ed. Kyushu Univ. 11 (1978), 135-143.

16. Saxena, R.K. and Pogány, T.K., Mathieu-type Series for the ℵ-function occurring in Fokker-Planck Equation. EJPAM, Vol. 3, No. 6 (2010): 980-988.

17. Saxena, R.K. and Pogány, T.K., On fractional integration formulae for Aleph functions, Appl. Math. Comput., 218 (2011): 985-990.

18. Saxena, V.P., Formal solution of certain new pair of dual integral equations involving H-function, Proc. Nat. Acad. Sci. India Sect.A., 51 (1982), 366-375.

19. Srivastava, H. M., A contour integral involving Fox’s H-function, Indian J. Math., 14, 1-6 (1972).

20. Südland, N., Baunqann, B. and Nonnenmacher, T.F., Open problem: Who knows about the Aleph (χ )-function? Fract. Cal. Appl. Anal. 1(4) (1998), 401-402.

21. Szego, C., Orthogonal polynomials, Amer. Math. Colloq. Publ. 23, 4th ed., Amer. Math. Soc. Providence, Rhode Island, 1975.

Source of support: Nil, Conflict of interest: None Declared.