/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 8, No. 2, 2015, 271-282

ISSN 1307-5543 – www.ejpam.com

Fractional Generalization of Rodrigues-type Formulas For

Certain Class Of Special Functions

Maged G. Bin-Saad

Department of Mathematics, University of Aden, Khormaksar P. O. Box 6014, Aden, Yemen

Abstract. This paper refers to some generalizations of certain classical Rodrigues formulas. By means

of the Riemann - Liouville operator of fractional calculus general Rodrigues-type representation for-

mulas of fractional order are derived and some of their properties are given and compared with the

corresponding properties of known cases.

2010 Mathematics Subject Classifications: 33C45, 26A33

Key Words and Phrases: Riemann-Liouville fractional differentiation and integration operators, Ro-

drigues’ formula, Laguerre, Hermite, Bessel and Humbert polynomials

1. Preliminaries and Definitions

The subject of fractional calculus is one of the most intensively developing areas of mathe-

matical analysis, mainly due to its fields of application range from biology through physics and

electrochemistry to economics, probability theory and statistics (see [8, 10, 12, 14]). Indeed,

on behalf of the nature of their definitions the fractional derivatives and integrals provide an

excellent instrument for the modeling of memory and hereditary properties of various ma-

terials and processes. Half-order derivatives and integrals prove to be more useful for the

formulation of certain electrochemical problems than the classical methods [1]. In this work,

based upon Riemann - Liouville fractional derivative and integral operators we introduce a

new generalized Rodrigues-type representation for a certain class of special functions involv-

ing Laguerre, Hermite , Bessel and Humbert polynomials, which provide further generalization

of a number of known Rodrigues - type formulas and new fractional Rodirgues-type formulas

(see [2–4, 9, 13]).

Let L1(I) be a class of Lebesque integrable functions on the interval I = [a, b] where

0 ≤ a < b <∞, and let Γ(·) be the gamma function. According to the Riemann-Liouville ap-

proach to fractional calculus the fractional derivative Dα of order α ∈ (n−1, n), (n= 1,2,3, . . .)

Email address: mgbinsaad@yahoo.com

http://www.ejpam.com 271 c© 2015 EJPAM All rights reserved.



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 272

of the function f (x) is given by (see[8, 12])

Dαa f (x) = In−α
a Dn f (x), D =

d

d x
, (1)

and the fractional integral of the function f (t) of order β is defined by (see [6–13])

I
β
t f (x) =

1

Γ(β)

∫ x

a

(x − s)β−1 f (s)ds. (2)

In comparison to the classical calculus let us mention that, for example, if µ ≥ 0, t > 0 and

α > −1, then the fractional derivative of the power function xα is given by

Dµxα =
Γ(α+ 1)

Γ(α−µ+ 1)
xα−µ. (3)

Definition 1. Let ν,γ ∈ (n− 1, n), n = 1,2,3, . . ., a, b,β ∈ ℜ and k = 1,2,3, . . .. We define the

generalized fractional Rodrigues formula by the two functions

F (β ,γ)
ν (a, b, k; x) =

x−β eaxk

Γ(ν+ 1)
Y (β ,γ)
ν (a, b, k; x), (4)

F
(β ,−γ)
−ν (a, b, k; x) =

x−β eaxk

Γ(1− ν)
Y
(β ,−γ)
−ν (a, b, k; x), (5)

where

Y (β ,γ)
ν (a, b, k; x) = Dνxβ+bγe−axk

, β + bγ > −1, (6)

and

Y
(β ,−γ)
−ν (a, b, k; x) = Iνxβ−bγe−axk

, β − bγ > −1. (7)

It is important to note that the Laguerre polynomials L
β
ν (x) and L

β
−ν(x) due to El-Sayed

[2, p.10, (5) and (6)], the Rodirgues formulas L
β
ν (γ, a; x) and L

β
−ν(−γ, a; x) due to Rida and

El-Sayed [13, p.30, (3) and (4)] and the Laguerre polynomials L(α)ν (x) introduced recently in

[8] and used by El-Sayed [3, p.10, (5)and (6)] and Mirevski (see [9, p.1273, (15)]) are special

cases of our formulas (4) and (5) as given below:

Γ(ν+ 1)

n!
F (β ,γ)
ν (1,

n

γ
, 1; x) =

x−β ex

n!
Dνxβ+ne−x = Lβν (x), (8)

Γ(1− ν)
n!

F
(β ,γ)
−ν (1,

n

γ
, 1; x) =

x−β ex

n!
Iνxβ−ne−x = L

β
−ν(x), (9)

F (β ,γ)
ν (a, 1, 1; x) =

x−β eax

Γ(ν+ 1)
Dνxβ+γe−ax = Lβν (γ, a; x), (10)

F
(β ,γ)
−ν (a, 1, 1; x) =

x−β eax

Γ(1− ν)
Iνxβ−γe−ax = L

β
−ν(−γ, a; x), (11)

F (α,ν)
ν (1,1,1; x) =

x−αex

Γ(ν+ 1)
Dνxα+νe−x = L(α)ν (x), (12)



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 273

F
(α,ν)
−ν (1,1,1; x) =

x−αex

Γ(1− ν)
Iνxα−νe−x = L

(α)
−ν (x), (13)

where the last formula is new and suggested by the assertion (5).

Next, the introduction of the Rodrigues formulas (4) and (5) leads us to generalization of

many well-known Rodrigues formulas up to fractional forms. In this regard the Rodrigues rep-

resentations (4) and (5), in particular, yield the following new fractional Rodrigues–type repre-

sentations for the generalized Hermite polynomials H(r)n (x , a, b) [5], the generalized Laguerre

polynomials L(α)n (x , k, p) [15], Bessel polynomials yn(x) and Humbert polynomials hn(x) as

follows:

(−1)νF (β ,γ)
ν (a, 0, k; x) =

(−1)νx−β e−axk

Γ(ν+ 1)
Dνxβ e−axk

= H(k)ν (x ,β , a), (14)

(−1)νF
(β ,γ)
−ν (a, 0, k; x) =

(−1)νx−β e−axk

Γ(1− ν)
Iνxβ e−axk

= H
(k)
−ν(x ,β , a), (15)

F (β ,ν)
ν (a, 1, k; x) =

x−β e−axk

Γ(ν+ 1)
Dνxβ+νe−axk

= Lβν (x , k, a), (16)

F
(β ,ν)
−ν (a, 1, k; x) =

x−β e−axk

Γ(1− ν)
Iνxβ−νe−axk

= L
β
−ν(x , k, a), (17)

a−νF (β−2,ν)
ν (a, 2,−1; x) =

a−νx−β+2e
a
x

Γ(ν+ 1)
Dνxβ+2ν−2+e

−a
x = yν(x ,β , a), (18)

a−νF
(β−2,ν)
−ν (a, 2,−1; x) =

a−νx−β+2e
a
x

Γ(1− ν)
Iνxβ−2ν−2+e

−a
x = y−ν(x ,β , a), (19)

F (0,ν)
ν (1,1,2; x) =

ex2

Γ(ν+ 1)
Dνxνe−x2

= hν(x), (20)

F
(0,ν)
−ν (1,1,2; x) =

ex2

Γ(1− ν)
Iνx−νe−x2

= h−ν(x). (21)

From the properties of the fractional calculus and the definitions (6) and (7), we can easily

prove the following lemma:

Lemma 1. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . .; a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

DαY (β ,γ)
ν (a, b, k; x) =Y

(β ,γ)
ν+α (a, b, k; x) = DνY (β ,γ)

α (a, b, k; x), (22)

DαY
(β ,γ)
−ν (a, b, k; x) =IνY (β ,γ)

α (a, b, k; x), (23)

DαY
(β ,−γ)
−ν (a, b, k; x) =IνY (β ,−γ)

α (a, b, k; x). (24)



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 274

2. Hypergeometric Series Representations

Taking to account that the generalized hypergeometric function pFq is defined by [17, p.19,

(2)]:

pFq

�

a1, . . . , ap; b1, . . . , bq; x
�

=

∞
∑

k=0

(a1)k . . . (ap)k

(b1)k . . . (bq)k

xk

k!
, (25)

and △(k;λ) abbreviates the array of k parameters λk , λ+1
k , . . . , λ+k−1

k , k = 1,2,3, . . ., we estab-

lish the following series representations for the functions Y
(β ,γ)
ν (a, b, k; x) and Y

(β ,−γ)
−ν (a, b, k; x).

Theorem 1. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . .; a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

Y (β ,γ)
ν (a, b, k; x) =

n
∑

p

�

n

p

�

Γ(β + bγ+ 1)Γ(β + bγ− n+ 1)xβ+bγ−ν

Γ(p− n+ 1)Γ(β + bγ− p+ 1)Γ(β + bγ− ν+ 1)

× 2kF2k

�

△(k; 1),△(k;β + bγ− n);△(k;β + bγ− ν+ 1),△(k; p− n+ 1);−axk
�

,

(26)

Y
(β ,−γ)
−ν (a, b, k; x) =xν+β−bγ Γ(β − bγ+ 1)

Γ(β − bγ+ ν+ 1)

× kFk

�

△(k;β − bγ+ 1);△(k;β − bγ+ ν+ 1);−axk
�

. (27)

Proof. From properties of the fractional calculus and the definition of Y
(β ,γ)
ν (a, b, k; x), we

get

Y (β ,γ)
ν (a, b, k; x) =In−ν

(

n
∑

p=0

�

n

p

�

�

Dp xβ+bγ
�

�

Dn−pe−axk
�

)

=

n
∑

p=0

�

n

p

� ∞
∑

q=0

(−a)qΓ(β + bγ+ 1)Γ(kq+ 1)

q!Γ(β + bγ− p+ 1)Γ(kq+ p− n+ 1)
In−ν

�

xβ+bγ+kq−n
�

. (28)

On putting n− ν= α, we get

Iα
�

xβ+bγ+kq−n
�

=
1

Γ(α)

∫ x

0

(x − s)α−1sβ+bγ+kq−nds,

which on putting x − s = x t, gives us

Iα
�

xβ+bγ+kq−n
�

=
xα+β+bγ+kq−n

Γ(α)

∫ 1

0

tα−1(1− t)β+bγ+kq−nd t.

Hence

In−ν = xβ+bγ+kq−ν Γ(β + bγ+ kq− n+ 1)

Γ(β + bγ+ kq− ν+ 1)
. (29)

Now, substituting from (29) into (28), we get

Y (β ,γ)
ν (a, b, k; x) =xβ+bγ−ν

n
∑

p=0

�

n

p

�



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 275

×
∞
∑

q=0

Γ(β + bγ+ 1)Γ(kq+ 1)Γ(β + bγ+ kq− n+ 1)

Γ(β + bγ− p+ 1)Γ(kq+ p− n+ 1)Γ(β + bγ+ kq− ν+ 1)

(−axk)q

q!
,

(30)

which on applying the results (see[17, p.16-17]):

(λ)n =
Γ(λ+ n)

Γ(λ)
, (31)

and

(λ)mn = mmnΠm
j=1

�

λ+ j − 1

m

�

n

, n= 0,1,2,3, . . . . (32)

yields the assertion (26). Similarly, we have

Y
(β ,−γ)
−ν (a, b, k; x) =

∞
∑

q=0

(−a)q

q!
Iν
�

xβ−bγ+kq
�

=

∞
∑

q=0

(−a)q

q!

1

Γ(ν)

∫ x

0

(x − s)ν−1sβ−bγ+kqds

=xβ−bγ+ν
∞
∑

q=0

(−axk)q

q!

1

Γ(ν)

∫ 1

0

tν−1(1− t)β−bγ+kqd t.

Thus

Y
(β ,γ)
−ν (a, b, k; x) = xβ−bγ+ν

∞
∑

q=0

(−axk)q

q!

Γ(β − bγ+ kq+ 1)

Γ(β − bγ+ kq+ ν+ 1)
, (33)

which on using (31) and (32) gives us the assertion (27) and this complete the proof of the

Theorem 1.

In the same manner one can easily prove the following useful result.

Theorem 2. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . ., a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

Y (β ,γ)
ν (a, b, k; x) =xβ+bγ−ν Γ(β + bγ+ 1)

Γ(β + bγ− ν+ 1)

× kFk

�

△(k;β + bγ+ 1);△(k;β + bγ− ν+ 1);−axk
�

. (34)

Proof. We infer to the proof of Theorem 1.

The following results are an immediate consequence of Theorems 1 and 2, respectively.

Corollary 1. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . ., a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

F (β ,γ)
ν (a, b, k; x) =

eaxk

Γ(ν+ 1)

n
∑

p

�

n

p

�

Γ(β + bγ+ 1)Γ(β + bγ− n+ 1)x bγ−ν

Γ(p− n+ 1)Γ(β + bγ− p+ 1)Γ(β + bγ− ν+ 1)



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 276

× 2kF2k

�

△(k; 1),△(k;β + bγ− n);△(k;β + bγ− ν+ 1),△(k; p− n+ 1);−axk
�

,

(35)

F
(β ,−γ)
−ν (a, b, k; x) =

eaxk

Γ(1− ν)
xν−bγ Γ(β − bγ+ 1)

Γ(β − bγ+ ν+ 1)

× kFk

�

△(k;β − bγ+ 1);△(k;β − bγ+ ν+ 1);−axk
�

. (36)

Corollary 2. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . ., a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

F (β ,γ)
ν (a, b, k; x) =

eaxk

x bγ−ν

Γ(ν+ 1)

Γ(β + bγ+ 1)

Γ(β + bγ− ν+ 1)

× kFk

�

△(k;β + bγ+ 1);△(k;β + bγ− ν+ 1);−axk
�

. (37)

3. Recurrence and Fractional Operational Relations

First, we establish the following pure recurrence relations.

Theorem 3. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . ., a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

Y (β ,γ)
ν (a, b, k; x) =(β + bγ)Y

(β−1,γ)

ν−1
(a, b, k; x)− akY

(β+k−1,γ)

ν−1
(a, b, k; x), (38)

Y
(β ,γ)

ν+1
(a, b, k; x) =DY (β ,γ)

ν (a, b, k; x)

=(β + bγ)Y (β−1,γ)
ν (a, b, k; x)− akY

(β+k−1,γ)

ν−1
(a, b, k; x), (39)

xY (β ,γ)
ν (a, b, k; x) =Y (β+1,γ)

ν (a, b, k; x)− νY
(β ,γ)

ν−1
(a, b, k; x), (40)

xY (β ,γ)
ν (a, b, k; x) =(β + bγ− ν+ 1)Y

(β ,γ)

ν−1
(a, b, k; x)− akY

(β+k,γ)

ν−1
(a, b, k; x). (41)

Proof.

• From (6), we have

Y (β ,γ)
ν (a, b, k; x) =Dν−1

�

(β + bγ)xβ+bγ−1e−axk

− akxβ+bγ+k−1e−axk
�

,

=(β + bγ)Dν−1
�

xβ+bγ−1e−axk
�

− akDν−1
�

xβ+bγ+k−1e−axk
�

,

which on using (6) gives us (38).

• Again, from (6), we have

DY (β ,γ)
ν (a, b, k; x) =Y

(β ,γ)

ν+1
(a, b, k; x) = Dν

�

(β + bγ)xβ+bγ−1e−axk

− akxβ+bγ+k−1e−axk
�

,

=(β + bγ)Dν
�

xβ+bγ−1e−axk
�

− akDν
�

xβ+bγ+k−1e−axk
�

,

which on using (6) gives us (39).



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 277

In view of the definition (6), if we expand the exponential function e−axk

in power series

and use (3), we obtain

Y (β ,γ)
ν (a, b, k; x) =

∞
∑

q=0

(−a)q

q!

Γ(β + bγ+ kq+ 1)

Γ(β + bγ+ kq− ν+ 1)
xβ+bγ−ν+kq, (42)

from which, we can prove the assertions (40) and (41).

Next, according to the formulas (4) to (7) it may of interest to point out that the polyno-

mials F
(β ,γ)
ν (a, b, k; x) and F

(β ,−γ)
−ν (a, b, k; x) have the following basic properties.

Theorem 4. Let ν,γ ∈ (n− 1, n), n= 1,2,3, . . . ; a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

DF (β ,γ)
ν (a, b, k; x) =(ν+ 1)F

(β ,γ)

ν+1
(a, b, k; x) +

�

akxk − β
x

�

F (β ,γ)
ν (a, b, k; x), (43)

(ν+ 1)F
(β ,γ)

ν+1
(a, b, k; x) =

�

β + bγ
�

F (β−1,γ)
ν (a, b, k; x)− akF (β+k−1,γ)

ν (a, b, k; x), (44)

νF (β ,γ)
ν (a, b, k; x) =

�

β + bγ
�

F
(β−1,γ)

ν−1
(a, b, k; x)− akF

(β+k−1,γ)

ν−1
(a, b, k; x), (45)

(1− ν)F (β ,γ)

1−ν (a, b, k; x) =
�

β + bγ
�

F
(β−1,γ)
−ν (a, b, k; x)− akF

(β+k−1,γ)
−ν (a, b, k; x), (46)

(1− ν)F (β ,γ)

1−ν (a, b, k; x) =(β + bγ)F
(β ,γ− 1

b )

−ν (a, b, k; x)− akF
(β+k−1,γ)
−ν (a, b, k; x), (47)

(1− ν)F (β ,γ)

1−ν (a, b, k; x) =DF
(β ,γ)
−ν (a, b, k; x) +

�

akxk − β
x

�

F
(β ,γ)
−ν (a, b, k; x), (48)

F (β ,γ+1)
ν (a, b, k; x) =x bF (β+b,γ)

ν (a, b, k; x) = x−bF (β−b,γ+2)
ν (a, b, k; x), (49)

F (β ,1−γ)
ν (a, b, k; x) =x bF (β+b,−γ)

ν (a, b, k; x) = x−bF (β−b,2−γ)
ν (a, b, k; x). (50)

Proof.

• From (4), we have

DF (β ,γ)
ν (a, b, k; x) =

−β
x

�

x−β eaxk

Γ(ν+ 1)
Dν
�

xβ+bγe−axk
�

�

+ akxk−1

�

x−β eaxk

Γ(ν+ 1)
Dν
�

xβ+bγe−axk
�

�

+ (ν+ 1)

�

x−β eaxk

Γ(ν+ 2)
Dν+1

�

xβ+bγe−axk
�

�

,

which gives us (43).

• From (4), we have

F
(β ,γ)

ν+1
(a, b, k; x) =

x−β eaxk

(ν+ 1)Γ(ν+ 1)
Dν
�

(β + bγ)xβ+bγ−1e−axk

− akxβ+bγ+k−1e−axk
�

,

which on using (4) gives us (44).



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 278

• We have

F (β ,γ)
ν (a, b, k; x) =

x−β eaxk

Γ(ν+ 1)
Dν−1

�

(β + bγ)xβ+bγ−1e−axk

− akxβ+bγ+k−1e−axk
�

,

which on using (4) gives us (45).

• We have

F
(β ,γ)

1−ν (a, b, k; x) =
x−β eaxk

Γ(2− ν)
D1−ν

�

xβ+bγe−axk
�

=
x−β eaxk

Γ(2− ν)
Iν
�

(β + bγ)xβ+bγ−1e−axk

− akxβ+bγ+k−1e−axk
�

,

which gives us (46).

• From (31) and the fact that

F
(β−1,γ)
−ν (a, b, k; x) = F

(β ,γ− 1
b )

−ν (a, b, k; x),

we get (47).

The proofs of the assertions (48) to (50) are similar to that of (43) to (47) , then we skip the

details.

Theorem 5. Let ν,γ ∈ (n− 1, n), n= 2,3, . . ., a, b,β ∈ ℜ and k = 1,2,3, . . .. Then

xβ e−axk
∞
∑

s=0

�

ν

s

�

Γ(ν− s+ 1)F
(β ,γ)
ν−s (a, b, k; x)

�

Ds f (x)
�

=In−v

�

xβ+bγe−axk

�

D+
β + bγ− akxk

x

�n

f (x)

�

, (51)

xβ e−axk
∞
∑

s=0

�

ν

s

�

Γ(ν− s+ 1)F
(β ,γ)
ν−s (a, b, k; x)

�

Ds f (x)
�

=In−v
¦

Ω
n,m

β ,γ,b
(x)

�

D− akxk−1
�m

f (x)
©

, (52)

where

Ω
n,m

β ,γ,b
(x) =

n
∑

m=0

�

n

n−m

��

β + bγ

n−m

�

(n−m)!eaxk

xβ+bγ+m−n,

and

xβ e−axk
∞
∑

s=0

�

ν

s

�

Γ(ν− s+ 1)F
(β ,γ)
ν−s (a, b, k; x)

�

Ds f (x)
�

=In−v

(

xβ+bγ−ne−axk
n−1
∏

j=0

�

x D+ β + bγ− j − akxk
�

f (x)

)

. (53)



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 279

Proof. By the generalized Leibnitz rule for fractional derivative [8, p.90, (4.3)], we find

Dν
�

xβ+bγe−axk

f (x)
�

=

∞
∑

s=0

�

ν

s

�

Dν−s
�

xβ+bγe−axk
�

Ds
�

f (x)
�

,

which in view of (4), gives us

Dν
�

xβ+bγe−axk
�

=

∞
∑

s=0

�

ν

s

�

Γ(ν− s+ 1)xβ e−axk

F
(β ,γ)
ν−s (a, b, k; x)Ds

�

f (x)
�

. (54)

On other hand, we obtain

Dν
�

xβ+bγe−axk

f (x)
�

= In−ν
�

Dn
�

xβ+bγe−axk

f (x)
��

,

which on using the shift relation [16]

Dn
�

eφ(x) f (x)
�

= eφ(x)
�

D+ Dφ(x)
�

f (x),

gives us

Dν
�

xβ+bγe−axk

f (x)
�

= In−ν

�

xβ+bγe−ax x

�

D+
β + bγ− akx x

x

�n

f (x)

�

. (55)

Hence from (54) and (55), we get (51). Similarly, since

Dν
�

xβ+bγe−axk

f (x)
�

= In−ν
�

Dn
�

xβ+bγe−axk

f (x)
��

=Iν−n

¨

n
∑

s=0

�

n

s

�

Ds
�

xβ+bγ
�

Dn−s
�

e−axk

f (x)
�

«

,

we find that

Dν
�

xβ+bγe−axk

f (x)
�

=In−v

¨

n
∑

m=0

�

n

n−m

��

β + bγ

n−m

�

(n−m)!eaxk

xβ+bγ+m−n
�

D− akxk−1
�m

f (x)

«

. (56)

Hence, from (54) and (56), we get (52). Next,we have

Dν
�

xβ+bγe−axk

f (x)
�

= In−ν
�

Dn−1
�

D
�

xβ+bγe−axk

f (x)
���

. (57)

Now

Dn−1
�

D
�

xβ+bγe−axk

f (x)
��

= Dn−1
�

xβ+bγe−axk �

x D+ β + bγ− akxk
�

f (x)
�

.

Let
�

x D+ β + bγ− akxk
�

f (x) = f1(x), then

Dn−1
�

xβ+bγe−axk

f1(x)
�

= Dn−2
�

xβ+bγe−axk �

x D+ β + bγ− 1− akxk
�

f1(x)
�



M. Bin-Saad / Eur. J. Pure Appl. Math, 8 (2015), 271-282 280

=Dn−2
�

xβ+bγe−axk �

x D+ β + bγ− 1− akxk
� �

x D+ β + bγ− akxk
�

f (x)
�

,

which on repetition of the process gives

Dn−1
�

D
�

xβ+bγe−axk

f (x)
��

=

 

xβ+bγ−ne−axk
n
∏

j=1

�

x D+ β + bγ+ j − n+ 1− akxk
�

f (x)

!

,

(58)

where the product has been taken in the operative sense. Since the operators involved are

commutative (60) can be written in the form

Dn−1
�

D
�

xβ+bγe−axk

f (x)
��

=

 

xβ+bγ−ne−axk
n−1
∏

j=0

�

x D+ β + bγ− j − akxk
�

f (x)

!

which with the help of (54) yields (53).

When f (x) = 1, assertions (51) to (53) reduce to the interesting fractional relations

F (β ,γ)
ν (a, b, k; x) =

x−β eaxk

Γ(ν+ 1)
In−v

�

xβ+bγe−axk

�

D+
β + bγ− akxk

x

�n�

. (59)

F (β ,γ)
ν (a, b, k; x) =

x−β eaxk

Γ(ν+ 1)
In−v

¦

Ω
n,m

β ,γ,b
(x)

�

D− akxk−1
�m
©

, (60)

and

F (β ,γ)
ν (a, b, k; x) =

x−β eaxk

Γ(ν+ 1)
In−v

(

xβ+bγ−ne−axk
n−1
∏

j=0

�

x D+ β + bγ− j − akxk
�

)

, (61)

respectively.

These are three fractional formulas which happens to give many new fractional represen-

tations for the special functions mentioned in the first section of this work as particular cases.

For example

Lβν (γ, a; x) =
x−β eax

Γ(ν+ 1)
In−v

�

xβ+γe−ax

�

D+
β + γ− ax

x

�n
�

, (62)

Lβν (γ, a; x) =
x−β eax

Γ(ν+ 1)
In−v

(

xβ+bγ−ne−ax

n−1
∏

j=0

�

x D+ β + bγ− j − ax
�

)

, (63)

H(k)ν (x ,β , a) =
(−1)νx−β eaxk

Γ(ν+ 1)
In−v

�

xβ e−axk

�

D+
β − akxk

x

�n�

, (64)

Lβν (x , k, a) =
x−β eaxk

Γ(ν+ 1)
In−v

�

xβ+νe−axk

�

D+
β + ν− akxk

x

�n�

, (65)

and

hν(x) =
ex2

Γ(ν+ 1)
In−v

�

xνe−x2

�

D+
ν− a2x2

x

�n�

. (66)



REFERENCES 281

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