On New Prajapati-Shukla Functions And...

Mathematics Today Vol.27(Dec-2011) 24-39 ISSN 0976-3228

On New Prajapati-Shukla Functions And Polynomials

J. C. Prajapati1

and A. K. Shukla2

1Department of Mathematical Sciences,

Faculty of Technology and Engineering,

Charotar University of Science and Technology,

Changa, Anand-388 421, India.

2Department of Applied Mathematics and Humanities,

S.V.National Institute of Technology,

Surat-395 007, India

E-mails: [email protected]

[email protected]

ABSTRACT

The principal aim of the paper is to introduce the various generalizations of

Shukla-Prajapati functions and polynomials. For these new functions and

polynomials their various properties including usual differentiation and

integration, Integral transforms, Generalised hypergeometric series form, Mellin

– Barnes integral representation, Recurrence relations, Integral representation,

Decomposition, Fractional calculus operators properties, generating relations,

bilateral generating relation and finite summation formulae of new class of

polynomials also established.

Key Words: Mittag–Leffler function, Integral Transforms, Fractional integral and differential

operators.

AMS classification(2000): 33E12, 33C20, 44A20, 30E20.

mailto:[email protected]

J. Prajapati & A. Shukla - On New Prajapati-Shukla Functions And Polynomials 25

1. INTRODUCTION AND PRELIMINARIES

Recently, Shukla and Prajapati (2007), investigated and studied the function

)(,

, zE q which is defined for ,, C; 0Re,0Re,0Re and )1,0(q N

as:

!)(

0

,

,n

z

nzE

n

n

qnq

, (1.1)

where )(

)()(

qnqn denotes the generalized Pochhammer symbol Rainville(1960)

which in particular reduces to

q

r n

qn

q

rq

1

1 if q N.

In continuation of the study, the generalization of (1) can be written as )(,

, zE which

is defined for 0Re,0Re,0Re and 0 as

!)(

0

,

,n

z

nzE

n

n

n

. (1.2)

This is a generalization of the exponential function exp(z), the confluent hypergeometric

function z;, Rainville (1960), the Mittag – Leffler function Mittag-Leffler (1903),

the Wiman’s function Wiman (1905) and the function )(, zE defined by Prabhakar (1971)

as well as equation (1.1).

Gorenflo et al. (1998), Gorenflo and Mainardi (2000), Kilbas et al. (1996, 2004), Saigo and

Kilbas (1998), Srivastava and Tomovski (2009), Tomovski et al (in press) and many other

researchers also studied the various properties of Mittag-Leffler functions and its

generalizations with their applications. The applications of Integral Transforms discussed by

Sneddon (1979)

The function )(,

, zE is an entire function of order 1

1Re

if 1Re , and

absolutely convergent in 1, RRz if 1Re .

The truncated power series of the function )(,

, zE can be defined as,

N

m

N

m

n

j n

jmn

jmn

m

nm

mnN

nm

z

mm

zzE

0

1

0

1

1

,,

,1

1)(

)(

!)(

)()(

(1.3)

and a special case for the study of )(,, zEq for

n

1 as:

0 0

1

1

,

,1

1)(

)(

!)(

)()(

m m

n

j n

jmn

jmn

m

nm

m

nm

z

mm

zzE

(1.4)

where 2n , 1N ; 0Re , 0Re and 0 .

26 Mathematics Today Vol.27(Dec-2011) 24-39

Authors investigated the operator fE aw

,

;,, as,

)()(])([)()()( ,

,

1,

;,, axdttftxwEtxxfE

x

a

aw

. (1.5)

where w C ; 0Re,0Re,0Re ; 0 .

If 1 , then (1.5) reduces to the result Prabhakar (1971).

Shukla and Prajapati (2008 A) introduced a class of polynomials which are connected by

Mittag-Leffler function )(xE , in continuation of the study of class of polynomials, Authors

introduced a general class of polynomial defined as,

xpExxpEn

xskaxA k

n

k

na

n

,

,

,

,

,,,

!,,; (1.6)

where ,,, are real or complex numbers; ska ,, are constants and generalized Mittag-

Leffler function defined as (1.2).

The proofs of all results established in this paper are parallel to Shukla and Prajapati (2007 A,

2007 B, 2007 C, 2008 A, 2008 B, 2009 A, 2009 B, 2010).

2. BASIC PROPORTIES OF THE FUNCTION )(,

, zE

As a consequence of the definitions (1.2) the following results hold:

THEOREM (2.1). If ,, C; 0Re,0Re,0Re and 0 then

)()()( ,

1,

,

1,

,

1, zEdz

dzzEzE

(2.1)

0

1,1

,

,

,!

)()(n

n

n

nn

zzzEzE

(2.2)

in particular,

)()()( ,1

,, zzEzEzE

. (2.3)

THEOREM (2.2). If w,,, C; 0Re,0Re,0Re and 0 then

for m N,

zEzEdz

d m

mm

m

,

,

,

,

(2.4)

zEzwzEzdz

d

m

mm

,

,

1,

,

1

, 0Re m (2.5)


in particular,

zEzwzEzdz

dm

mm

,1

,1 (2.6)

and

wzmz

mwzz

m

dz

d m;,;,

11

. (2.7)

THEOREM (2.3). If r

p with rp, N relatively prime; , C and 0 then

)(,

,

r

p

r

pp

p

zEdz

d

!

1

1

1

1 n

pr

n

z

pr

np

r

np

r

npn

n

, (2.8)

in particular,

r

n

zr

n

zezE

r

n

r

r 1

,1

)(

1

1

1

1

, (r = 2, 3 . . .) . (2.9)

3. GENERALIZED HYPERGEOMETRIC FUNCTION REPRESENTATION OF

)(,, zEq

Using (1.2), taking k N and N then we have

;;

;;1,,

z

FzEk

. (3.1)

Convergence criteria for generalized hypergeometric function kF :

(i) If k , the function kF converges for all finite z.

(ii) If 1 k , the function kF converges for 1z and diverges for 1z .

(iii) If 1 k , the function kF divergent for 0z .

(iv) If 1 k , the function kF is absolutely convergent on the circle 1z if

.011

Re1 1

k

j i

i

k

j


where ; is a -tuples

1,...,

1,

;

; is a -tuples

1,...,

1,

.

4. MELLIN-BARNES INTEGRAL REPRESENTATION OF )(,

, zE

THEOREM (4.1). Let ,;R C ( 0 ) and N. The function )(,

, zE is

represented by the Mellin – Barnes integral as:

dszs

qss

izE s

L

)(

)(

)()(

)(2

1)(,

,

, (4.1)

where zarg ; the contour of integration beginning at i and ending at , i and

indented to separate the poles of the integrand at ns for all n N 0 (to the left) from

those at

q

ns

for all n N 0 (to the right).

5. INTEGRAL TRANSFORMS OF )(,

, zE

In this section, some useful integral transforms like Euler transforms, Laplace transforms,

Mellin transforms and Whittaker transforms also discussed.

THEOREM (5.1). Euler (Beta) transforms

1

0;,,,

;,,,)1(

22

,

,

11

ba

xabdzxzEzz ba

, (5.1)

where 0Re,0Re,0Re,0Re,0Re,0Re ba and 0 .

THEOREM (5.2). Laplace transforms.

0 ;,

;,,,

12,

,

1

s

xaasdzxzEez za s

, (5.2)

where 0Re,0Re,0Re,0Re,0Re a , 0 and .1s

x


THEOREM (5.3). Mellin transforms.

0

,

,

1 )( dtwtEt s

sw

sss

, (5.3)

where 0Re,0Re,0Re,0Re s and 0 .

To obtain Whittaker transform, we use the following integral,

01

2

1

2

1

,12

v

vv

dttWte vt

, where 2

1Re v . (5.4)

THEOREM (5.4). Whittaker Transforms.

0

,

,,21

1dtwtEptWet

pt

;,1,,

;,2

1,,

23

p

wqp

, (5.5)

where 0Re,0Re,0Re,0Re,0Re and 0 .

6. RECURRENCE RELATIONS

THEOREM (6.1). For any Re 0 p , Re 0 s and Re 0 , 0 we get

zEzE spsp

,

2,

,

1,

zEzspp

zEzpzEss

sp

spsp

,

3,

,

3,

22,

3,

12

2

(6.1)

where )]([)( ,

,

,

, zEdz

dzE qq

and )]([)( ,

,2

2,

, zEdz

dzE qq

.

THEOREM (6.2). For k and m N

)()()2()()]1(2[)()( ,

2,

,

3,

,

3,

,

3,

22,

1, zEzEmmzEmkkzzEzkzE mkmkmkmkmk

.

(6.2)

7. INTEGRAL REPRESENTATIONS

THEOREM (7.1). For any Re 0 p , Re 0 s and Re 0 , 0 and setting

kp , ms , where k and m N then

1

0

,

, )( dttEt k

mk

m = )1()1( ,

2,

,

1,

mkmk EE (7.1)


THEOREM (7.2). If 0Re,0Re,0Re,0Re and 0 then

dttEtzt

z

0

,

,

11 )()()(

1

= )(,

,

1

zEz

. (7.2)

Special cases of Theorem (7.2): For 0Re , from (7.2), the particular cases listed as below:

dtetz

z

t

0

1)()(

1

= )(1,1

1,1 zEz

, (7.3)

dtttz

z

0

1 )cosh()()(

1

= )( 21,1

1,2 zEz

, (7.4)

dtt

ttz

z

0

1)sinh(

)()(

1

= )( 21,1

2,2 zEz

, (7.5)

dtzerfztz c

z

)(exp)()(

1

0

21

= )(1,1

1,2

1 zEz

. (7.6)

THEOREM (7.3). If 0Re,0Re,0Re and 0 then

dttEtzt

z

0

2,

,2

11 )()()(

1

= )]()([ 2,

,2

,

,

1

zEzEz . (7.7)


0

1,

,4 )(

2

dxxxEe t

x

=

2,

2

1,

2

2

tEt . (7.8)


)()()()( ,

,

,

1,

2,

,

1

axEaxExaxExdx

d

. (7.9)


)(1 ,

,

1

,

1

ztEts

zsL

. (7.10)


THEOREM (7.7). If 0Re,0Re,0Re , 0 and 0 then

0

1

0

,

,!)(

)(exp)(

n k

n

n

n

k

k

dtnn

tt

z

tzkzE

. (7.11)

dtztEtzE qq )()1()(

1)( ,

,

1

0

1

1

,

,

. (7.12)

dttzEttzE qq

)1()1(

)(

1)( ,

,

1

0

11,

,

. (7.13)

THEOREM (7.8). If 0Re,0Re,0Re,0Re,0Re,0Re , C

and

0 then

)()()1()(

1 ,

,

1

0

,

,

11 zEduzuEuu

. (7.14)

)()()()()()(

1 ,

,

1,

,

11 txEtxdstsEtssx

x

t

. (7.15)

If 1 then

wxExdtwtEtxwEtxt v

v

x

v

v 1,

,

1

0

1,

,

1,

,

11 )()()(

. (7.16)

wzEzdtwtEt

z

,

1,

0

,

,

1

. (7.17)

in particular,

wzEzdtwtEt

z

1,

0

,

1

(7.18)

and );1,();,(0

1 wzz

dtwtt

z

. (7.19)


8. DECOMPOSITION OF MITTAG-LEFFLER FUNCTION

THEOREM (8.1). Integral Representations of the function )(,

,1 zEn

)(,

,1 zEn

= )(,

,1

nzE +

duuuzEn kn

n

k

z

nn

nk

11

1 0

,

,1 )(1

1

, (8.1)

where 2n ; 0Re , 0Re and 0 .

THEOREM (8.2). If 2n , 0Re , 0Re and 0 then

n

n

zEzdz

d1

,

,1

1

1

1

)(1

1

)(1

!)(

)(

!)(

)( n

k nm

n

m

nmn

k nk

n

k

kn

nm

z

kn

zz

(8.2)

THEOREM (8.3). )(,

,1 zEn

n

nk

kn

k

zEz

Re1

,

,1

1

0

, (8.3)

where 2n ; 0Re , 0Re and 0 .

Remark of Theorem 8.3: It is easy to verify that,

zEzE nN

n

nN

n

,,

,1

,,

,1 )(

zE

n

,

,1

n

nk

kn

k

zEz

,

,1

1

0 1

. (8.4)

THEOREM (8.4). If 2n ; 0Re , 0Re and 0 then

)()( .,

,1

,

,1 zEzE nN

nn

T (8.5)

where

nn

k nnNk

nNk

N

Nn

NzE

z

NNN

zT Re

1)(

)(

!)1()1(

)(,

,1

1

1

1

)1(

. (8.6)

9. FRACTIONAL INTEGRAL AND DIFFERENTIAL OPERATORS

ASSOCIATEDWITH THE FUNCTION )(,

, zE

The following well-known facts are prepared for studying properties of the Riemann-

Liouville fractional integrals and differential operators associated with the function )(,

, zE q

and also the properties of operator fE aw

,

;,, .


• ),( baL Space of Lebesgue measurable real or complex valued functions Kilbas et

al.(2004):

),( baL Consists of Lebesgue measurable real or complex valued functions )(xf on ],[ ba

i.e. ),( baL =

b

a

dttfff )(:1

. (9.1)

Kilbas et al. (2004) studied the several properties of fractional integral and differential

integral

operators.

• Confluent hypergeometric functions Rainville (1960):

This is also known as the Pochhammer – Barnes confluent hypergeometric function defined

as,

0

11!)(

)();;();,(

n n

n

n

nb

zazbaFzba , (9.2)

where 0b or a negative integer is convergent for all finite z.

• Gauss multiplication theorem Rainville (1960):

If m is a positive integer and z C then,

)()2(1

2

1

2

1

1

mzmm

kz

mzmm

k

. (9.3)

• Riemann-Liouville fractional integrals of order Khan and Abukhammash GS (2003)

Let ),()( baLxf , C )0)(Re( then

)(xfI xa

= )(xfIa

= dt

tx

tfx

a

1)(

)(

)(

1 ( ax ) (9.4)

is called R-L left-sided fractional integral of order .

Let ),()( baLxf , C ( 0)Re( ) then

)(xfIbx

= )(xfIb

= dt

xt

tfb

x

1)(

)(

)(

1 ( bx ) (9.5)

is called R-L right-sided fractional integral of order .


Theorem (9.1). Let ),0[ Ra and let w C; 0)Re(),Re(),Re(),Re( ,

0 for ax , then

)(})({)[( ,

,

1 xatwEatIa

= ])([)( ,

,

1

axwEax

(9.6)

and

)(})({)[( ,

,

1 xatwEatDa

= ])([)( ,

,

1

axwEax

. (9.7)

Theorem (9.2). Let ,, C and 0 then

)]([ ,1

1,0

xEI x = )(,1

1,

xEx (9.8)

in particular,

)()( 1,10 xExeI x

x

. (9.9)


0 for ax , then

)()( 1,

;,, xatE q

aw

=

)()()( ,

,

1 axwEax q

. (9.10)


0 for ab then the operator q

awE ,

;,,

is bounded on ),( baL and

11

,

;,, fBfE q

aw

(9.11)

where !

)(

)]Re()[Re()(

)()(

)Re(

0

)Re(

k

abw

kkabB

k

k

qk

(9.12)

the relation fEfEI q

aw

q

awa

,

;,,

,

;,,

(9.13)

hold for any summable function ),( baLf .

We can express the function )(,

, zEm

, m N as:

1

0 02

1

2

1

,

,

!

)(1)2()(

m

k nnm

n

n

n

m

mm

z

nm

k

m

km

zE

. (9.14)


On substituting 1 in (9.14) and then the result Kilbas et al. (2004) becomes a special case

of (9.14) as:

1

02

1

2

1

,

1,

, ;,1)2(

)()(m

km

m

mmm

z

m

k

m

km

zEzE

. (9.15)

10. FRACTIONAL OPERATORS AND FUNCTION )(,

, zE

Consider the function

0

2)!(

)()()(

n

n

n

n

cttf

, where 0)Re( , 0

and c is an arbitrary constant then the fractional integral operator of order can be written as,

)()( ,

1,1 ctEttfI

. (10.1)

We denote the function (10.1) as ),,,( qcEt , i.e. )(),,,( ,

1,1 ctEtcEt

. (10.2)

The fractional differential operator of order can be written as

02)!(

)()()(

n

n

nkn

n

ctIDtfD

)(,

1,1 ctEt

. (10.3)

We denote the function (10.3) as ),,,( qcEt , i.e. )(),,,( ,

1,1 ctEtcEt

(10.4)

THEOREM (10.1). If 0)Re( , 0 , c is an arbitrary constant and fractional integral

operator of order then

),,,(),,,( cEcEI tt . (10.5)

),,,(),,,( cEcED tt . (10.6)

The Laplace transforms of ),,,( cEt is given as

,

11

1)},,,({

s

c

scEL t , (10.7)

In the light of Theorem (10.1), we can prove following Theorem (10.2).

THEOREM (10.2). If 0)Re( , 0 , c is an arbitrary constant and fractional integral

operator of order then

),,,(),,,( cEqcEI tt . (10.8)


),,,(),,,( cEqcED tt . (10.9)

,

11

1)},,,({

s

c

scEL t . (10.10)

11. GENERATING RELATIONS AND FINITE SUMMATION FORMULAE OF (6)

A considerably large number of special functions (including all of the classical orthogonal

polynomials) are known to possess generating relations.

We used operational technique by employing Mittal (1977), Patil and Thakare (1975) as a

differential operator, where )( xDsxa and )1(1 xDxa , dx

dD , for obtaining

following generating relations and finite summation formulae of (1.6).

0

),,,( ),,;(n

n

n tskaxA = }])1({[)}({)1(

1,

,

,,

akk

as

atxpExpEat

. (11.1)

!),,;(

0

),1,1,1(

0 m

tskaxA

a

u m

m

km

n

n

n

0

1

),1,1,1( ,,;n

nk

k

k

n

t

a

uska

txAe

. (11.2)

0

),,,( ),,;(n

nan

n tskaxA = }])1({[)}({)1(

1,

,

,,

1 akk

as

atxpExpEat

. (11.3)

0

),,,(

)( ),,;(m

m

nm tskaxAm

mn

=

},,;)1(

}])1({[

)}({)1(

1{

),,,(

1

,

,

,

,skaatxA

atxpE

xpEat a

n

a

k

kasn

. (11.4)

0

),,,(

)( ),,;(n

nan

nm tskaxAm

mn

=

},,;)1(

}])1({[

)}({)1(

1{

),,,(

1

,

,

,

,1skaatxA

atxpE

xpEat a

m

a

k

kas

. (11.5)


Using (11.1) to (11.5), we obtained following generating relations

0

),,,(

)( ),,;(m

m

nm

n zskaxAm =

}])1({[

)}({)1(

1

,

,

,

,

a

k

k

atxpE

xpEaz a

s

m

am

n

m az

zskaazxAmnSm

1

],,;)1([),(!

1

),,,(

0

(11.6)

)0;;(1

0

aazNn

0

),,,(

)( ),,;(m

mam

nm

n zskaxAm =

}])1({[

)}({)1(

1

,

,

,

,1

a

k

k

atxpE

xpEaz a

s

m

aam

m

n

m az

zskaazxAmnSm

1

],,;)1([),(!

1

),,,(

0

. (11.7)

)0;;(1

0

aazNn

Two finite summation formulae for (1.6) also obtained as

),,;(!

1),,;( )0,,,(

)(

0

),,,( skaxAa

am

skaxA mn

m

n

n

m

n

. (11.8)

),,;(!

1),,;( ),,,(

)(

0

),,,( skaxAa

am

skaxA mn

m

n

n

m

n

. (11.9)

we get following bilateral generating relation for ),,;(),,,( skaxAqn

,

0

,

),,,(

)( )(),,;(m

mb

mm tyRskaxA

=

],)1([

}])1({[

)}({)1(

1

,1

,

,

,

, ba

b

a

k

kasn

ytatx

atxpE

xpEat

, (11.10)

where

0

),,,(

)(,, ),,;(],[m

m

bmmb tskaxAtx , 0, m


and )(, yRb

m is a polynomial of degree

b

m in y, which is defined as

)(, yRb

m =

bm

k

k

k ybk

m/

0

,

; b is a positive integer, is an arbitrary complex number.

REFERENCES

[1] Carlitz L. (1977): “Some expansion and convolution formulas related to Mac Mohan’s

master theorems”, SIAM Journal of Mathematical Analysis, 8 (2), 320 – 336.

[2] Gorenflo R, Kilbas A.A. and Rogosin S.V. (1998): “On the generalised Mittag-Leffler

type function”, Integral Transforms and Special Functions, 7, 215-224.

[3] Gorenflo R. and Mainardi F. (2000): “On Mittag – Leffler function in fractional

evaluation processes”, Journal of Computational and Applied Mathematics, 118, 283 –

299.

[4] Khan M.A. and Abukhammash G.S. (2003): “A study on two variable analogues of

certain fractional operators”, Pro Mathematica, 17(33), 32-48.

[5] Kilbas A.A. and Saigo M. (1996): “On Mittag – Leffler type function, fractional

calculus operators and solution of Integral equations”, Integral Transforms and Special

Functions, 4, 355 – 370.

[6] Kilbas A.A., Saigo M. and Saxena R.K. (2004): “Generalised Mittag-Leffler function

and generalised Fractional calculus operators”, Integral Transforms and Special

Functions, 15, 31-49.

[7] Mittag – Leffler G.M. (1903) : “Sur la nouvelle fonction xE ”, C.R. Acad. Sci. Paris,

137, 554 – 558.

[8] Mittal H.B. (1977): “Bilinear and bilateral generating relations”, American Journal of

Mathematics , 99, 23-55.

[9] Patil K.R. and Thakare N.K. (1975): “Operational formulas for a function defined by a

generalized Rodrigues formula-II”, Science Journal of Shivaji University, 15, 1-10.

[10] Prabhakar T.R. (1971): “A singular integral equation with a generalized Mittag-Leffler

function in the Kernel”, Yokohama Mathematical Journal, 19, 7-15.

[11] Prajapati J.C. (2008). “Generalization of Mittag-Leffler function and its applications”,

Ph.D. thesis, Department of Mathematics, Sardar Vallabhbhai National Institute of

Technology, Surat, India

[12] Rainville E.D. (1960): “Special Functions” The Macmillan Company, New York.


[13] Saigo M. and Kilbas A.A. (1998). “On Mittag – Leffler type function and

applications”, Integral Transforms and Special Functions, 7, 97-112.

[14] Sneddon I.N. (1979): “The use of Integral Transforms”, Tata McGraw – Hill Pub. Co.

Ltd., New Delhi.

[15] Shukla A.K. and Prajapati J.C. (2007 A): “On a generalization of Mittag-Leffler

function and its properties”, Journal of Mathematical Analysis and Applications, 337,

pp. 797-811.

[16] Shukla A.K. and Prajapati J.C. (2007 B): “Some properties of a class of Polynomials

suggested by Mittal”, Proyecciones Journal of Mathematics, 26(2), pp. 145-156.

[17] Shukla A.K. and Prajapati J.C. (2007 C). “Generalization of a class of Polynomials”,

Demonstratio Mathematica, 40 (4), 819-826.

[18] Shukla AK and Prajapati JC (2008 A). “A general class of polynomials associated with

generalized Mittag-Leffler function. Integral Transforms and Special Functions 19(1),

pp. 23-34.

[19] Shukla AK and Prajapati JC (2008 B). “On Generalized Mittag-Leffler type function

and

generated integral operator”, Mathematical Sciences Research Journal 12 (12), pp.

283-290.

[20] Shukla A.K. and Prajapati J.C. (2009 A): “Some remarks on generalized Mittag-Leffler

function”, Proyecciones Journal of Mathematics 28(1), 27-34.

[21] Shukla A.K. and Prajapati J.C. (2009 B): “On a Recurrence relation of generalized

Mittag-Leffler function. Surveys in Mathematics and its Applications, 4, 133-138.

[22] Shukla A.K. and Prajapati J.C. (2010): “Decomposition and properties of generalized

Mittag-Leffler function”, Communicated for publication.

[23] Srivastava H.M. and Tomovski Z. (2009): “Fractional calculus with an integral

operator containing a generalized Mittag-Leffler function in the kernel”, Applied

Mathematics and Computation, 211, 198–210.

[24] Tomovski Z., Hilfer R. and Srivastava H.M. (2010): “Fractional and operational

calculus with generalized fractional derivative operators and Mittag-Leffler type

functions”, Integral Transforms and Special Functions, 21(11), 797-814.

[25] Wiman A. (1905): Uber de Fundamental satz in der Theorie der Funktionen xE ”,

Acta Mathematica, 29, 191 – 201.

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