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ON GENERALIZED MITTAG-LEFLER TYPE FUNCTIONS

Rudolf Gorenflo, Anatoly A.Kilbas and Sergei V.Rogosin

Department of Mathematics and Informatics, Freie Universitat Berlin,Arnimallee 2-6, 14195 Berlin, Germany

Department od Mathematics and Mechanics, Belarusian State University,Fr Skaryny Ave 4, 220050 Minsk, Belarus

Abstracts

The paper is devoted to study the properties of the special functions

E.m.l(z) =

k=0

ckzk

, z C,

c0= 1, ck =k1i=0

([im+l] + 1)

([im+l+ 1] + 1) (k= 1, 2, )

with >0, m >0, real l and (im+l) + 1 = 0, 1, 2, for i = 0, 1, 2, .For m = 1, E,1,l(z) coincide with the Mittag-Leffler type function

E,l+1(z) =k=0

zk

(k+l+ 1)

up to the constant (l + 1). The order and type of the entire function

E.m.l(z) are evaluated and recurrent relations are given. For the functionsEn,m,l(z) (n = 1, 2, ) connections with functions of hypergeometric type arestudied and differentiation formulae are proved.

1. Introduction.

Recently it turned out the role of the well-known function

E(z) =k=0

zk

(k+ 1) ( C, Re()> 0) (1.1)

introduced by Mitttag-Leffler [29] and titled his name [9]. The properties of thisentire function investigated by many mathematicians, can be found in [1],[6]-[7],

[9], [12]-[14], [30], [37]-[38], etc. A number of applied problems were found forwhich the knowledge of properties ofE(z) are of great importance. Some ofthese problems were studied in [3]-[5], [10]-[11], [15], [25], [27], [31], [35]. Mainlythese applications are due to connections of Mittag-Leffler function (1.1) withfractional calculus [6], [14], [17], [20]-[24], [28], [33], and in particular in solutions

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of integral equations and differential equations of fractional order [10], [16]-[17],

[22]-[24], [28], [32], [35].At the beginning of the 50th a certain generalization of the function (1.1)

E,(z) =k=0

zk

(k+) (, C, Re()> 0, Re()> 0), (1.2)

called Mittag-Leffler type function [6]-[7], was introduced by Agarwal and Hum-bert [2], [18], [19]. The properties of (1.2) (in particular asymptotic behaviorand zero distributions) were studied in [6], [7], [9], [12], [34], [36].

A new generalization of (1.2) was given in [20] in connection with solutionof Abel-Volterra integral equations. Such a function is defined by

E,m,l(z) =

k=0

ckzk (z C)ck =

k1

i=1

([im+l] + 1)

([im+l+ 1] + 1)

(k= 0, 1, 2, )

(1.3)where an empty product is recognized equal to one, and , m and l are realnumbers such that

>0, m > 0, (im+l) + 1=1, 2, 3, (i= 0, 1, 2, ). (1.4)

In particular, ifm = 1, the conditions in (1.4) take the form

>0, (i+l) + 1=1, 2, 3, (i= 0, 1, 2, ). (1.5)

and (1.3) is reduced to the Mittag-Leffler type function given in (1.2):

E,1,l(z) = (l+ 1)E,l+1(z). (1.5)

Therefore we shall call E,m,l(z) generalized Mittag-Leffler type function. Thisfunction was useful to solve in closed form of new classes of integral equationsand differential equations of fractional order [21]-[24], [32].

When = n N ={1, 2, }, En,m,l(z) takes the form

En,m,l(z) = 1 +k=1

k1i=0

1

[n(im+l) + 1] [n(im+l) +n]

zk, (1.6)

wheren, mand l are real numbers such that

n N, m > 0, n(im+l)=1, 2, , n (i= 0, 1, 2, ). (1.7)

Our paper is devoted to study the properties of the generalzed Mittag-Lefflertype functionsE,m,l(z) andEn,m,l(z) given by (1.3) and (1.6). Section 2 deals

with evaluating the order and type of an entire function E,m,l(z)( > 0).Section 3 is devoted to recurrent formulas for such a function. In section 4 weshow that functions En,m,l(z) (n = 1, 2, ) are functions of hypergeometrictype. Differentiation properties ofEn.m.l(z

nm) andEn.m.l(znm) are studied in

Section 5.

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2. Order and type of the entire function E,m,l(z).

In this section we find some characteristics ofE.m.l(z). First of all we showthat the generalized Mittag-Leffler type function (1.3) is an entire function.

Lemma 1. If, m and l are real numbers such that the condition (1.4) issatisfied, thenE,m,l(z) is an entire function of the variablez.

Proof. According to (1.3) and the relation [8, 1.18(4)]

(z+a)

(z+b) =zab

1 +

(a b)(a+b 1)

2z +O

1

z2

(z ), (2.1)

we have the asymptotic estimate

cn+1

cn

= [(nm+l) + 1]

[(nm+l+ 1) + 1]

(mn) 0 (n ).

Therefore by DAlembert criterion the power series in (1.2) converges in thewhole complex plane, and hence E,m,l(z) is an entire function.

Corollary 1. For > 0, m > 0 and l > 1/ E,m,l(z) is an entirefunction ofz.

Corollary 2. If n N, m > 0 and l are real numbers such that thecondition (1.7) is satisfied, then the generalized Mirrag-Leffler type functionEn,m,l(z) given by (1.6) is an entire function ofz.

The main characteristics of an entire functions are its order and type.Definition 1. [26] An entire functionF(z) is said be to be a function of

finite order if there exists a positive constantk such that the inequality

max|z|=r

|F(z)|< exp(rk) (2.2)

is valid for all sufficiently large values of r (r > r0(k)). The greatest lowerbound of such numbersk is called the order of the entire functionF(z):

= inf{k >0 : (2.1) takes place }. (2.3)

For a function of a given order a more precise characterization of the growthis given by the type of the function.

Definition 2. [26] By the type of an entire functionF(z) of order wedenote the greatest lower bound of positive numbersA for which asmptotically

max|z|=r

|F(z)|< exp(Ar) : (2.4)

= inf{a > 0 : (2.3) takes place }. (2.5)

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If the entire function F(z) can be represented in the form of power series

F(z) =

n=0

cnzn, (2.6)

being convergent for all z C, then the order and type ofF(z) are evaluatedby [26]

= limsupn

n log(n)

log(1/|cn|), (2.7)

(e)1/ = limsupn

n1/(|cn|)

1/n

. (2.8)

The order and type of the generalized Mittag-Leffler type function (1.3) isgiven by the following statement.

Theorem 1. If, m and l are real numbers such that the condition (1.4)is satisfied, then E,m,l(z) is an entire function of order = 1/ and type= 1/m. Moreover the following asymptotic estimate holds

|E,m,l(z)|< exp

1

m+

|z|1/

, |z| r0 > 0, (2.9)

whenever >0 is sufficiently small.Proof. Applying (2.7) we first find the order ofE,m,l(z). According to

(1.3) we have

cn= (l+ 1)(l+m+ 1) (l+m[n 1] + 1)

(l++ 1)(l++m+ 1) (l++m[n 1] + 1).

Letzk = l+mk+ 1 (k N ={1, 2, }). (2.10)

Using (2.1) with z = zk, a = 0 and b= we obtain that for any d > 0 thereexistk0 N such that

(1 d)zk

(zk+)(zk) (1 +d)zk k > k0. (2.11)

Therefore forn > k0 we have

log

1

|cn|

=

n1

k=0log

(zk+)

(zk)

=

k0k=0

log

(zk+)(zk) +

n1k=k0+1

log

(zk+)(zk)

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d1+

n1k=k0+1

log(1 +d) + +

n1k=k0+1

log(zk )

=d1 + (n k0 +1)log(1+d) + n1

k=k0+1

log(k) + log(m) + log

1 +

l + 1

klm

and hence

log

1

|cn|

d3+nd4+n log(n), (2.12)

whered3 andd4 are certain positive constants. Similarly

log

1

|cn|

d4+ (n k0+ 1) log(1 d)

+n1

k=k0+1

log(k) + log(m) + log

1 +

l+ 1

klm

and

log

1

|cn|

d5+nd6+n log(n). (2.13)

It follows from (2.12) and (2.13) that the following usual limit exists

limn

n log(n)

log(1/|cn|)=

1

(2.14)

and hence in accordance with (2.7) order ofE,m,l(z) is given by

= 1

. (2.15)

Next we use (2.8) to find the type ofE,m,l(z). Applying (2.11), (2.10) and(1.3) we have

k0k=0

(zk+)(zk)

1

1 +c

nk01 n1k=k0+1

zk |cn|

k0k=0

(zk+)

(zk)

1

1 c

nk01 n1k=k0+1

zk . (2.16)

Using this formula and asymptotic relation

n1k=k0

1

km

1

n!

1

m

n e

nm

n(n ),

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from (2.8) we obtain =m and hence type ofE,m,l(z) is given by

= 1

m. (2.17)

The asymptotic estimate (2.9) follows from (2.7)-(2.8) and Definitions 1 and2. This completes the proof of theorem.

Corollary 1. If n N, m > 0 and l are real numbers such that thecondition (1.7) is satisfied, then the generalized Mirrag-Leffler type functionEn,m,l(z) given by (1.6) is an entire function ofz with order= 1/n and type= 1/m.

Corollary 2. The Mittag-Leffler functionE(z)and the Mittag-Leffler typefunctionE,(z) given in (1.1) and (1.2), have the same order and type:

= 1

, = 1. (2.18)

Remark 1. The assertions of Corollary 2 are well known - see [6], [7], [26].Remark 2. Theorem 1 shows that the generalized Mittag-Leffler type func-

tion (1.3) has the same order as Mittag-Leffler function (1.1) and Mittag-Lefflertype function (1.2). But the type ofE,m,l(z) depends onm.

3. Recurrent relations for E,m,l(z).

In this section we give recurrent relations for E,m,l(z).Theorem 2. Let, mandl be real numbers such that the condition (1.4) is

satisfied and letn N. Then for generalized Mittag-Leffler type function (1.3)the recurent relation

zn [E,m,l+nm(z) 1] =n1j=0

[(jm +l+ 1) + 1][(jm+l) + 1]

E,m,l(z) 1 n

i=1

i1

j=0

[(jm+l) + 1]

[(jm+l+ 1) + 1]

zi

(3.1)

is valid.Proof. By (1.3) we have

E,m,l+nm(z) = 1 +k=1

k1i=0

[(im+nm+l) + 1]

[(im+nm+l+ 1) + 1]

zk.

Making the changes of the order of summation j = i+ n and i = k+ n, weobtain

E,m,l+nm(z) = 1 +k=1

n+k1

j=n

[(jm+l) + 1]

[(jm+l+ 1) + 1]

zk

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= 1 +

i=n+1

i1j=n

[(jm +l) + 1]

[(jm +l+ 1) + 1]

zin

= 1 +n1j=0

[(jm+l+ 1) + 1]

[(jm+l) + 1]

i=n+1

i1

j=0

[(jm +l) + 1]

[(jm +l+ 1) + 1]

zin

= 1 + 1

zn

n1j=0

[(jm +l+ 1) + 1]

[(jm +l) + 1]

1 +

i=1

i1j=0

[(jm +l) + 1]

[(jm +l+ 1) + 1]

zi

n

i=1

i1

j=0

[(jm +l) + 1]

[(jm+l+ 1) + 1]

zi 1

= 1 + 1

zn

n1j=0

[(jm +l) + 1]

[(jm+l+ 1) + 1]E,m,l(z) n

i=1

i1

j=0

[(jm +l) + 1]

[(jm +l+ 1) + 1]

zi 1

and (3.1) is proved.Corollary 1. If the conditions of Theorem 2 are valid, then

zE,m,l+m(z) =(l++ 1)

(l+ 1) [E,m,l(z) 1] (3.2)

and

znE,m,l+nm(z) =n1j=0

[(jm+l+ 1) + 1]

[(jm+l) + 1]E,m,l(z) 1 n1

i=1

i1

j=0

[(jm +l) + 1]

[(jm +l+ 1) + 1]

zi

(3.3)

forn= 2, 3, .Corollary 2. If > 0, > 0 and n N, then for Mittag-Leffler type

functionE,n+(z) the recurrent relations hold

zE,+(z) = E,(z) 1

() (3.4)

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and

zn(n+)E,n+(z) =

=n1j=0

(j++)

(j+)

()E,(z) 1 n1

i=1

i1

j=0

(j+)

(j++)

zi

(3.5)

forn= 2, 3, .Corollary 3. If > 0 and n N, then Mittag-Leffler type function

E,n+1(z) is expressed via Mittag-Leffler function (1.1) by

zE,+1(z) = E(z) 1 (3.6)

and

znE,n+1(z) = E(z) n

i=1

zi

(i+ 1)

(3.7)

forn= 2, 3, .

4. Connection ofEn,m,l(z) with functions of hypergeometric type.

As we have mentioned in Introduction, E,m,l(z) is generalization of theMittag-Leffler type functionE,(z) in (1.2), in particular of the Mittag-LefflerfunctionE(z) in (1.1). When= n N,En,m,l(z) in (1.6) becomes a functionof hypergeometric type. Such a function pFq(a1, a2. , ap; b1, b2. , bq; z) for

p N0 = N

{0},q N0, a1, a2. , ap C, b1, b2, , bq C and z C, |z|< 1, is defined by the hypergeometric series [8]

pFq(a1, a2. , ap; b1, b2. , bq; z) =

k=0

(a1)k(a2)k (ap)k

(b1)k(b2)k (aq)k

zk

k! , (4.1)

where(a)0 = 1, (a)k = a(a+ 1) (a+k 1) (k= 1, 2, ). (4.2)

Theorem 3. Let the condition (1.7) be satisfied. Then En,m,l(z) is givenby

En,m,l(z) = 1Fn

1;

nl+ 1

nm ,

nl+ 2

nm

nl+n

nm ;

z

(nm)n

(4.3)

and it is an entire function ofz with order = 1/n and type = 1/m.Proof. According to (1.3), (4.1) and (4.2) we have

En,m,l(z) = 1 +

k=1

k1i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

zk

= 1 +k=1

k!

(nl+ 1) (n[(k 1)m+l]) (nl+n) (n[(k 1)m+l] +n)

zk

k!

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= 1 +

k=1

(1)k

([nl+ 1])/[nm])k ([nl+n]/[nm])k

1

(nm)nk

zk

k!

=1 Fn

1;

nl + 1

nm ,

nl+ 2

nm

nl+n

nm ;

z

(nm)n

and (3.3) is proved. The last assertion follows from Theorem 1. This completesthe proof.

Corollary 1. Ifn N and >0, then the Mittag-Leffler functionEn,(z)is given by

En,1,(1)/n(z) = ()En,(z)

= 1Fn

1;

nl+ 1

n ,

nl+ 2

n

nl +n

n ;

z

(n)n

(4.4)

and it is an entire function of order= 1/n and type = 1.Corollary 2. If

m > 0, l R, im+l=1, 2, 3, (i= 0, 1, 2, ), (4.5)

thenE1,m,l(z) is given by

E1,m,l(z) = 1F1

1;

l+ 1

m ;

z

m

=

l+ 1

m

E1,(l+1)/m

zm

(4.6)

and it is an entire function ofz with order = 1 and type = 1/m.Corollary 3. If l=1, 2, 3, , thenE1,1,l(z) is given by

E1,1,l(z) = 1F1(1; l+ 1; z) = (l+ 1)E1,l+1(z) (4.7)

and it is an entire function ofz with order = 1and type = 1. In particular,if l N0,

E1,1,l(z) = l!E1,l+1(z) = l!

zl

ez

l1k=0

zk

k!

(4.8)

andE1,1,0(z) = E1(z) = e

z. (4.9)

5. Differentiation properties ofEn,m,l(z).

In this section we give two differentiation formulae for the generalized Mittag-Leffler type function (1.3). The first such a relation is given by the followingstatement.

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Theorem 4. Letn N, m > 0 andl , R be numbers such that conditions

in (1.7) are satisfied and leta C (a= 0). Then forEn,m,l(aznm)the followingdifferentiation formula

d

dz

n zn(lm+1)En,m,l(az

nm)

=

nj=1

[n(lm)+j]zn(lm)+aznlEn,m,l(aznm)

(5.1)holds. In particular, if

n(l m) = j f or some j = 1, 2, , n, (5.2)

then

d

dzn

zn(lm+1)En,m,l(az

nm)

= znlEn,m,l(az

nm). (5.3)

Proof. By (1.3) we haved

dz

n zn(lm+1)En,m,l(az

nm)

=

d

dz

n zn(lm+1) + j=1

cjajzn(lm+1)+nmj

= [n(l m+ 1)][n(l m+ 1) 1] [n(l m+ 1) n+ 1]zn(lm)

+

j=1

j1

i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

[n{(j 1)m+l+ 1} + 1]

[n({(j 1)m+l} + 1] ajzn(lm)+nmj

=

nj=1

[n(l m) +j]zn(lm) +aznl

+j=2

j2i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

ajzn(lm)+nmj.

Taking the change of summation k = j 1, we obtaind

dz

n zn(lm+1)En,m,l(z

nm)

=

nj=1

[n(l m) +j]zn(lm)

+aznl

+k=1

k1i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

ak+1znl+nmk

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=

nj=1

[n(lm)+j]zn(lm)

+aznl

1 +

k=1

k1i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

(aznm

)k

and (5.1) is proved in accordance with (1.3). (5.3) follows from (5.1). Theoremis proved.

Corollary 1. If n = 1, 2, , > 0 and a C (a = 0), then for theMittag-Leffler type functionEn,(azn) in (1.2)

d

dz

n z1En,(az

n)

= 1

( n)zn1 +az1En,(az

n). (5.4)

Corollary 2. Ifn = 1, 2, , k N (1 k n) anda C (a= 0), thenfor the Mittag-Leffler type functionEn,k(az

n)d

dz

n

zk1En,k(azn)

= azk1En,k(zn). (5.5)

In particular whenk= 1, for the Mittag-Leffler functionEn(azn) in (1,1)

d

dz

n[En(az

n)] = aEn(azn). (5.6)

Remark 3. By (1.2), the relation (5.4) can be represented in the formd

dz

n z1En,(az

n)

= zn1En,n(azn), (5.7)

which is known for a = 1 [6, (1.6)].Remark 3. Whena = 1, the relation (5.6) is well known - see [9, 18.1(15)].Next differentiation relation for the generalized Mittag-Leffler type function

(1.3) is given byTheorem 5. Letn N, m > 0 andl , R be numbers such that conditions

in (1.7) are satisfied and let a C (a = 0). Then for En,m,l(aznm) the

following differentiation formulad

dz

n zn(ml)1En,m,l(az

nm)

=

n

j=1

[n(m l) j]zn(ml1)1 + (1)nazn(l+1)1En,m,l(aznm) (5.8)

holds. In particular, if the conditions in (5,2) are satisfied, thend

dz

n zn(ml)1En,m,l(az

nm)

= (1)nazn(l+1)1En,m,l(znm). (5.9)

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Proof. By (1.3) we haved

dz

n zn(ml)1)En,m,l(az

nm)

=

d

dz

n zn(ml)11 + j=1

cjajzn(ml)nmj1

= [n(m l) 1] [n(m l) n]zn(ml)1n

+

j=1

aj

j1i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

[n(m l) nmj 1] [n(m l) nmj n]zn(ml)nmjn1

=n

j=1

[n(m l) j]zn(ml1)1

+(1)nj=1

j1i=1

([im+l] + 1)

([im+l+ 1] + 1)

(n[{j 1}m+l+ 1] + 1)

(n[{j 1}m+l] + 1) ajzn(ml)nmjn1

=n

j=1

[n(m l) j]zn(ml1)1

+(1)nazn(l+1)1

1 +

j=2

aj

j2i=1

(n[im+l] + 1)

(n[im+l+ 1] + 1)

ajznm(j1)

.

Making the change of summation k = j 1, we obtaind

dz

n zn(ml)1En,m,l(az

nm)

=

nj=1

[n(m l) j]zn(ml1)1+

+(1)nazn(l+1)1

1 +

k=1

k1i=1

(n[im+l] + 1)(n[im+l+ 1] + 1)

(aznm)k

and (5.8) is proved in accordance with (1.3). (5.9) follows from (5.8). Theoremis proved.

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Corollary 1. If n = 1, 2, , > 0 and a C (a = 0), then for the

Mittag-Leffler type functionEn,(azn) in (1.2)d

dz

n znEn,(az

n)

= (1)n

( n)z + (1)naznEn,(az

n). (5.10)

Corollary 2. Ifn = 1, 2, , k N (1 k n) anda C (a= 0), thenfor the Mittag-Leffler type functionEn,k(az

n)d

dz

n znkEn,k(az

n)

= (1)naznkEn,k(azn). (5.11)

In particular whenk= 1, for the Mittag-Leffler functionEn(azn) in (1,1)

ddzn

zn1En(azn)

= (1)nazn1En(azn). (5.12)

Remark 5. The relations (5.1), (5.8) and (5.3), (5.9) can be consideredas nonhomogeneous and homogeneous dfferential equations of order n for thefunctionszn(lm+1)En,m,l(az

nm) andzn(ml)1En,m,l(aznm), respectively. In

this way explicit solutions of new classes of differential equations were obtainedin [24], [32]. In particular, (5.4), (5.10) and (5.5), (5.11) are nonhomogeneousand homogeneous differential equations for the functions z1En,(az

n) and[znjEn,(azn). The function zj

1En,(azn) as explicit solution of diferen-tial equation was found earlier by reduction of the differential equation to thecorresponding Volterra integral equation [33, Section 42.1].

AcknowledgmentThe work was fulfilled under partial financial support of Belarusian Fund of

Fundamental Scientific Research.

References

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