# On Generalisation of Polynomials in Complex Plane

Authors: Maslina Darus; Rabha W. Ibrahim
Journal: Advances in Decision Sciences
 (2010)

---

## Abstract

The generalised Bell and Laguerre polynomials of fractional-order in complexz-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.


---

## Body

## 1. Introduction and Preliminaries

Special functions play important roles in applied mathematics. It has been seen that these functions have appeared in different frameworks, such as the mathematical physics [1], the combinatorial analysis [2], and the statistics [3]. Indeed, the explicit relationships between special functions and generalised hypergeometric functions have been obtained and mentioned in [4, 5]. Some extension of these polynomials already appeared in literature (see [6, 7]), and generalisation by using different type of calculus such as q-deform calculus [8, 9] and fractional calculus [10] has been studied.Definition 1.1.
The Bell polynomials take the form [11]
(1.1)Bn(y)=e-yDney=∑d=1nBn,d,n∈N0,
where
(1.2)Bn,d=∑|k|=d,‖k‖=nn!k!(y11!)k1⋯(ynn!)kn,
such that |k|=k1+⋯+kn, k1≥0,…,kn≥0, k!=k1!⋯kn!, and B0=1.Definition 1.2.
The Laguerre polynomials take the form(1.3)Ln(x)=exDne-xxn,L0=1,n∈N0.
Recall that Bell polynomials and Laguerre polynomials are classical mathematical tools for representing the nth derivative of a composite functions. Moreover, the multidimensional polynomials of higher order are already defined, which are suitable to represent the derivative of a composite function of several variables (see [6]).In this paper, we introduce definitions for these polynomials of arbitrary order (fractional order) in complex plane.In [12] the definitions for fractional operators (derivative and integral) in the complex z-plane ℂ are given as follows.Definition 1.3.
The fractional derivative of orderα is defined, for a function f(z), by
(1.4)Dzαf(z):=1Γ(1-α)ddz∫0zf(ζ)(z-ζ)αdζ,0≤α<1,
where the function f(z) is analytic in simply-connected region of the complex z-plane ℂ containing the origin and the multiplicity of (z-ζ)-α is removed by requiring log(z-ζ) to be real when (z-ζ)>0. For α∈[n-1,n) and n=1,2,…,
(1.5)Dzαf(z)=1Γ(n-α)dndzn∫0zf(ζ)(z-ζ)αdζ.Definition 1.4.
The fractional integral of orderα is defined, for a function f(z), by
(1.6)Izαf(z):=1Γ(α)∫0zf(ζ)(z-ζ)α-1dζ,α≥0,
where the function f(z) is analytic in simply-connected region of the complex z-plane (ℂ) containing the origin and the multiplicity of (z-ζ)α-1 is removed by requiring log(z-ζ) to be real when (z-ζ)>0. Further details in fractional calculus can be found in [13].Remark 1.5.
From Definition1.3, we have Dz0f(z)=f(0), limα→0Izαf(z)=f(z), and limα→0Dz1-αf(z)=f′(z). Moreover,
(1.7)Dzα{zμ}=Γ(μ+1)Γ(μ-α+1){zμ-α},μ>-1,0≤α<1,Izα{zμ}=Γ(μ+1)Γ(μ+α+1){zμ+α},μ>-1,α≥0,z≠0.Lemma 1.6 (see [14]).
Forα∈[0,1) and f is a continuous function, then
(1.8)DIzαf(z)=(z)α-1Γ(α)f(0)+IzαDf(z);D=ddz.
By using the operators (1.5) and (1.6), we define generalised polynomials in complex z-plane.Definition 1.7.
Letα∈[n-1,n)  and  n=1,2,…. The generalised Bell polynomials of order α and −α are
(1.9)Bα(z)=e-zDzαez,(1.10)B-α(z)=e-zIzαez,
respectively.Definition 1.8.
Letα∈[n-1,n)  and  n=1,2,…. The generalised Laguerre polynomials of order α and −α are
(1.11)Lα(z)=ezDzαe-zzm,(1.12)L-α(z)=ezIzαe-zzm,m∈N0,
respectively.Our plan is as follows. In Section2, we study the recurrence relations of the polynomials (1.9)–(1.12), the other three sections, we introduce the Laguerre-type of some special functions.


## 2. Recurrence Relations

In this section, we introduce some recurrence relations for the generalised Bell polynomials and Laguerre polynomials.Theorem 2.1.
Letα∈[0,1). Then the generalised Bell polynomials of order α and −α satisfy
(2.1)(1)DBα(z)=Bα+1(z)-Bα(z),(2)DB-α(z)=B1-α(z)-B-α(z)=zα-1e-zΓ(α),
where D:=d/dz.Proof.
Letα∈[0,1), then we have
(2.2)(1)DBα(z)=D[e-zDzαez]=e-zD[Dzαez]-e-zDzαez=e-zDzα+1ez-e-zDzαez=Bα+1(z)-Bα(z),(2)DB-α(z)=D[e-zIzαez]=e-zD[Izαez]-e-zIzαez=e-zIzα-1ez-e-zIzαez=B1-α(z)-B-α(z).
On the other hand and in virtue of Lemma 1.6, we have
(2.3)DB-α(z)=D[e-zIzαez]=e-zD[Izαez]-e-zIzαez=e-z[(z)α-1Γ(α)+Izαez]-e-zIzαez=zα-1e-zΓ(α).Theorem 2.2.
Letα∈[0,1). Then the generalised Laguerre polynomials of order α and −α satisfy
(2.4)(1)DLα(z)=Lα+1(z)+Lα(z),(2)DL-α(z)=L1-α(z)+Lα(z),(3)DL-α(z)=mΓ(α)zm+α-1,z≠0,
where D:=d/dz.Proof.
Letα∈[0,1), then we have
(2.5)(1)DLα(z)=D[ezDzαe-zzm]=ezD[Dzαe-zzm]+ezDzαe-zzm=ez[Dzα+1e-zzm]+Lα(z)=Lα+1(z)+Lα(z),(2)DL-α(z)=D[ezIzαe-zzm]=ezD[Izαe-zzm]+ezIzαe-zzm=ez[Izα-1e-zzm]+Lα(z)=L1-α(z)+Lα(z).
For z≠0 and in view of Lemma 1.6, we have
(2.6)(3)DL-α(z)=D[ezIzαe-zzm]=ezD[Izαe-zzm]+ezIzαe-zzm=ez[IzαDe-zzm]+ezIzαe-zzm=mΓ(α)zm+α-1.In addition, we have the following results.Theorem 2.3.
Letα∈[0,1). Then the generalised Bell polynomials Bα(z) are univalent solutions for the ordinary differential equation
(2.7)D2Bα(z)+2DBα(z)+Bα(z)=ρα(z),z≠0,
where ρα(z):=(αe-zzα-1[(α+1)z-1-1]+e-zz-α+z-α)/Γ(1-α).Proof.
DifferentiatingDBα(z) in Theorem 2.1 (part 1), using the fact that Bα+1(z)=DBα(z)+Bα(z) and using the properties in Lemma 1.6, into it, we obtain the result. Now for z1≠0,z2≠0 such that z1≠z2 and by applying Remark 1.5 on (1.9), we can verify that Bα(z) are univalent functions.Theorem 2.4.
Letα∈[0,1). Then the generalised Laguerre polynomials Lα(z) are univalent solutions for the ordinary differential equation
(2.8)D2Lα(z)+2DLα(z)+Lα(z)=θα(z),z≠0,
where θα(z):=ez[(z-α/Γ(1-α))(me-zzm-1-e-zzm)]′′.Proof.
DifferentiatingDLα(z) in Theorem 2.2 (part 1), using the fact that Lα+1(z)=DLα(z)-Lα(z) and again using Lemma 1.6, into it, we obtain the result. Now for z1≠0,z2≠0 such that z1≠z2 and by applying Remark 1.5 on (1.11), we obtain that Lα(z) are univalent functions.


## 3. Laguerre-Type Mittag-Leffler Function

In this section, the fractional Laguerre-type derivatives𝒟L introduced and in connection with a fractional differential isomorphism denoted by the symbol 𝒟z-β, acting onto the space 𝒜 of analytic functions of the z variable given as follows:
(3.1)D:=ddz⟶DL,z⟶Dz-1,
where(3.2)Dz-1f(z)=∫0zf(ζ)dζ.
In general,(3.3)Dz-βf(z):=1Γ(β)∫0zf(ζ)(z-ζ)β-1dζ,β≥0,
so that(3.4)F0(zβ):=Dz-β(1)=∫0z(z-ζ)β-1dζ=zβΓ(β+1),Fn(zβ)=Dz-β(zn)=1Γ(β)∫0zζn(z-ζ)β-1dζ=Γ(n+1)Γ(n+β+1)zn+β.
According to this isomorphism, the Mittag-Leffler operator Eλ(z) (see [15])(3.5)Eλ(z):=∑n=0∞znΓ(λn+1),λ>0,
is transformed into the first Laguerre-type Eλ1(z)(3.6)F0(Eλ(z))=∑n=0∞F0(zn)Γ(λn+1)=∑n=0∞znΓ(n+1)Γ(λn+1):=Eλ1(z).
This result can be generalised by considering the k Laguerre-type Mittag-Leffler(3.7)F0k(Eλ(z))=∑n=0∞F0(zn)[Γ(λn+1)]k=∑n=0∞znΓ(n+1)[Γ(λn+1)]k:=Eλk(z).
Thus(3.8)DLkEλk(az)=aEλk(az),a∈C.
Note that when λ=1 this reduces to exponential function (see [4]).


## 4. Laguerre-Type Hypergeometric Function

We use the same method of the previous section to obtain the Laguerre-type hypergeometric function(4.1)qFp(α1,…,αq;β1,…,βp;z)=∑n=0∞(α1)n⋯(αq)n(β1)n⋯(βp)nznn!,
where (a)n is the Pochhammer symbol defined by(4.2)(a)n=Γ(a+n)Γ(a)={1,n=0,a(a+1)⋯(a+n-1),n={1,2,…}.
According to the previous definition of Laguerre fractional derivative, the hypergeometric function qFp(α1,…,αq;β1,…,βp;z) is transformed into the first Laguerre-type qFp1(α1,…,αq;β1,…,βp;z)(4.3)F0(qFp(α1,…,αq;β1,…,βp;z))=∑n=0∞(α1)n⋯(αq)n(β1)n⋯(βp)nF0(zn)Γ(n+1)=∑n=0∞(α1)n⋯(αq)n(β1)n⋯(βp)nzn[Γ(n+1)]2=qFp1(α1,…,αq;β1,…,βp;z).
For k order we have
(4.4)F0k(qFp(α1,…,αq;β1,…,βp;z))=∑n=0∞[(α1)n⋯(αq)n(β1)n⋯(βp)n]kF0(zn)[Γ(n+1)]k=∑n=0∞[(α1)n⋯(αq)n(β1)n⋯(βp)n]kzn[Γ(n+1)]k+1=qFpk(α1,…,αq;β1,…,βp;z)
the Laguerre-type hypergeometric function.


## 5. Laguerre-Type Fox-Wright Function

Lastly, we introduce the Laguerre-type Fox-Wright function by using the similar approach in Section3. For complex parameters(5.1)α1,…,αq(αjAj≠0,-1,-2,…;j=1,…,q),β1,…,βp(βjBj≠0,-1,-2,…;j=1,…,p),
We have the Fox-Wright generalisation qΨp[z] of the hypergeometric qFp function by (see [16–18])(5.2)qΨp[(α1,A1),…,(αq,Aq);(β1,B1),…,(βp,Bp);z]=qΨp[(αj,Aj)1,q;(βj,Bj)1,p;z]:=∑n=0∞Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)znn!=∑n=0∞∏j=1qΓ(αj+nAj)∏j=1pΓ(βj+nBj)znn!,
where Aj>0 for all j=1,…,q, Bj>0 for all j=1,…,p, and 1+∑j=1pBj-∑j=1qAj≥0 for suitable values |z|. The Laguerre-type qΨp[(αj,Aj)1,q;(βj,Bj)1,p;z] is(5.3)F0(qΨp[(αj,Aj)1,q;(βj,Bj)1,p;z])=∑n=0∞Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)F0(zn)Γ(n+1)=∑n=0∞Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)zn[Γ(n+1)]2=qΨp1[(αj,Aj)1,q;(βj,Bj)1,p;z].
For k order we have (5.4)F0k(qΨp[(αj,Aj)1,q;(βj,Bj)1,p;z])=∑n=0∞[Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)]kF0(zn)[Γ(n+1)]k=∑n=0∞[Γ(α1+nA1)⋯Γ(αq+nAq)Γ(β1+nB1)⋯Γ(βp+nBp)]kzn[Γ(n+1)]k+1=qΨpk[(αj,Aj)1,q;(βj,Bj)1,p;z],
the Laguerre-type Fox-Wright function.



---

*Source: 230184-2010-11-28.xml*