Research Article On Generalisation of Polynomials in Complex Plane

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1 Hindawi Publishing Corporation Advances in Decision Sciences Volume 21, Article ID 23184, 9 pages doi:1.1155/21/23184 Research Article On Generalisation of Polynomials in Complex Plane Maslina Darus and Rabha W. Ibrahim School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 436 Selangor Darul Ehsan, Malaysia Correspondence should be addressed to Maslina Darus, maslina@ukm.my and Rabha W. Ibrahim, rabhaibrahim@yahoo.com Received 29 May 21; Accepted 28 September 21 Academic Editor: Shelton Peiris Copyright q 21 M. Darus and R. W. Ibrahim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The generalised Bell and Laguerre polynomials of fractional-order in complex z-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. 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 1 has been studied. Definition 1.1. The Bell polynomials take the form 11 n B n y e y D n e y B n,d, n N, 1.1 d1

2 2 Advances in Decision Sciences where B n,d k d, k n n! k! k1 kn yn, 1.2 1! n! y1 such that k k 1 k n, k 1,...,k n, k! k 1! k n!, and B 1. Definition 1.2. The Laguerre polynomials take the form L n x e x D n e x x n, L 1, n N. 1.3 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 order α is defined, for a function fz,by Dzfz α 1 d : Γ1 α dz fζ α dζ, z ζ α<1, 1.4 where the function fz 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 logz ζ to be real when z ζ >. For α n 1,n and n 1, 2,..., d n Dzfz α 1 Γn α dz n fζ z ζ α dζ. 1.5 Definition 1.4. The fractional integral of order α is defined, for a function fz, by I α z fz : 1 Γα fζz ζ α 1 dζ, α, 1.6 where the function fz 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 logz ζ to be real when z ζ >. Further details in fractional calculus can be found in 13.

3 Advances in Decision Sciences 3 Remark 1.5. From Definition 1.3, we have Dzfz f, lim α Iz α fz fz, and lim α Dz 1 α fz f z. Moreover, D α z{z μ } Γ μ 1 Γ μ α 1 { z μ α}, μ > 1, α<1, Iz α {z μ Γ μ 1 } Γ μ α 1 {zμα }, μ > 1, α, z/. 1.7 Lemma 1.6 see 14. For α, 1 and f is a continuous function, then DI α z fz zα 1 Γα f Iα z Dfz; D d dz. 1.8 By using the operators 1.5 and 1.6, we define generalised polynomials in complex z-plane. Definition 1.7. Let α n 1,n and n 1, 2,... The generalised Bell polynomials of order α and α are B α z e z D α ze z, B α z e z I α z e z, respectively. Definition 1.8. Let α n 1,n and n 1, 2,... The generalised Laguerre polynomials of order α and α are L α z e z D α ze z z m, 1.11 L α z e z I α z e z z m, m N, 1.12 respectively. Our plan is as follows. In Section 2, we study the recurrence relations of the polynomials , 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.

4 4 Advances in Decision Sciences Theorem 2.1. Let α, 1. Then the generalised Bell polynomials of order α and α satisfy 1 2 DB α z B α1 z B α z, DB α z B 1 α z B α z zα 1 e z Γα, 2.1 where D : d/dz. Proof. Let α, 1, then we have 1 DB α z D [ e z D α ze z] e z DD α ze z e z D α ze z e z Dz α1 e z e z Dze α z B α1 z B α z, 2 DB α z D [ e z Iz α e z] 2.2 e z DI α z e z e z I α z e z e z Iz α 1 e z e z Iz α e z B 1 α z B α z. On the other hand and in virtue of Lemma 1.6, we have DB α z D [ e z I α z e z] e z DIz α e z e z Iz α e z [ ] z α 1 e z Γα Iα z e z e z Iz α e z 2.3 zα 1 e z Γα. Theorem 2.2. Let α, 1. Then the generalised Laguerre polynomials of order α and α satisfy 1 DL α z L α1 z L α z, 2 DL α z L 1 α z L α z, 3 DL α z m Γα zmα 1, z/, 2.4 where D : d/dz.

5 Advances in Decision Sciences 5 Proof. Let α, 1, then we have 1 DL α z D [ e z D α ze z z m] e z D [ D α ze z z m] e z D α ze z z m e z[ Dz α1 e z z m] L α z L α1 z L α z, 2 DL α z D [ e z Iz α e z z m] 2.5 e z D [ I α z e z z m] e z I α z e z z m e z[ Iz α 1 e z z m] L α z L 1 α z L α z. For z / and in view of Lemma 1.6, we have 3 DL α z D [ e z I α z e z z m] e z D [ I α z e z z m] e z I α z e z z m e z[ I α z De z z m] e z I α z e z z m m Γα zmα In addition, we have the following results. Theorem 2.3. Let α, 1. Then the generalised Bell polynomials B α z are univalent solutions for the ordinary differential equation D 2 B α z 2DB α z B α z ρ α z, z/, 2.7 where ρ α z : αe z z α 1 α 1z 1 1e z z α z α /Γ1 α. Proof. Differentiating DB α z in Theorem 2.1 part 1, using the fact that B α1 z DB α z B α z and using the properties in Lemma 1.6, into it, we obtain the result. Now for z 1 /,z 2 / such that z 1 / z 2 and by applying Remark 1.5 on 1.9, we can verify that B α z are univalent functions. Theorem 2.4. Let α, 1. Then the generalised Laguerre polynomials L α z are univalent solutions for the ordinary differential equation D 2 L α z 2DL α z L α z θ α z, z/, 2.8 where θ α z : e z z α /Γ1 αme z z m 1 e z z m.

6 6 Advances in Decision Sciences Proof. Differentiating DL α z in Theorem 2.2 part 1, using the fact that L α1 z DL α z L α z and again using Lemma 1.6, into it, we obtain the result. Now for z 1 /,z 2 / such that z 1 / z 2 and by applying Remark 1.5 on 1.11, weobtainthatl α z are univalent functions. 3. Laguerre-Type Mittag-Leffler Function In this section, the fractional Laguerre-type derivatives D L introduced and in connection with a fractional differential isomorphism denoted by the symbol D β z, acting onto the space A of analytic functions of the z variable given as follows: D : d dz D L, z D 1 z, 3.1 where D 1 z fz fζdζ. 3.2 In general, D β z fz : 1 Γ β fζz ζ β 1 dζ, β, 3.3 so that F z β : D β z 1 F n z β D β z z n 1 Γ β z ζ β 1 dζ ζ n z ζ β 1 dζ z β Γ β 1, Γn 1 Γ n β 1 znβ. 3.4 According to this isomorphism, the Mittag-Leffler operator E λ z see 15 E λ z : z n, λ >, 3.5 Γλn 1 n is transformed into the first Laguerre-type E 1 λ z F E λ z F z n Γλn 1 z n Γn 1Γλn 1 : E1 λ z. n n 3.6 This result can be generalised by considering the k Laguerre-type Mittag-Leffler F k E F z n λz Γλn 1 z n k Γn 1Γλn 1 : k Ek λ z. n n 3.7

7 Advances in Decision Sciences 7 Thus D k L Ek λ az aek λaz, a C. 3.8 Note that when λ 1 this reduces to exponential function see Laguerre-Type Hypergeometric Function We use the same method of the previous section to obtain the Laguerre-type hypergeometric function qf p α1,...,α q ; β 1,...,β p ; z α 1 n α q n β1 n β p n n z n n!, 4.1 where a n is the Pochhammer symbol defined by a n Γa n Γa 1, n, aa 1 a n 1, n {1, 2,...}. 4.2 According to the previous definition of Laguerre fractional derivative, the hypergeometric function q F p α 1,...,α q ; β 1,...,β p ; z is transformed into the first Laguerre-type qf 1 pα 1,...,α q ; β 1,...,β p ; z F qf p α1,...,α q ; β 1,...,β p ; z α 1 n α q n β1 n β F z n p Γn 1 n α 1 n α q n β1 n β z n p Γn 1 2 n n n 4.3 q F 1 p α1,...,α q ; β 1,...,β p ; z. For k order we have F k qf p α1,...,α q ; β 1,...,β p ; z [ ] k α1 n α q n β1 n β F z n p Γn 1 k n [ ] k α1 n α q n n β1 n β z n p n Γn 1 k1 q Fp k α1,...,α q ; β 1,...,β p ; z n 4.4 the Laguerre-type hypergeometric function.

8 8 Advances in Decision Sciences 5. Laguerre-Type Fox-Wright Function Lastly, we introduce the Laguerre-type Fox-Wright function by using the similar approach in Section 3. For complex parameters α 1,...,α q β 1,...,β p αj /, 1, 2,...; j 1,...,q, A j βj /, 1, 2,...; j 1,...,p, B j 5.1 We have the Fox-Wright generalisation q Ψ p z of the hypergeometric q F p function by see [ α1,a 1,..., ] α q,a q ; [ qψ p z q Ψ p αj,a j 1,q β1,b 1,..., βp,b ; β j,b j 1,p ; z] p ; : Γα 1 na 1 Γ α q na q Γ zn β 1 nb 1 Γ βp nb p n! n q j1 Γ α j na j p j1 Γ zn β j nb j n!, n 5.2 where A j > for all j 1,...,q, B j > for all j 1,...,p,and1 p j1 B j q j1 A j for suitable values z. The Laguerre-type q Ψ p α j,a j 1,q ; β j,b j 1,p ; z is [ F qψ p αj,a j 1,q ; β j,b j 1,p ; z] Γα 1 na 1 Γ α q na q n Γ F z n β 1 nb 1 Γ βp nb p Γn 1 Γα 1 na 1 Γ α q na q n Γ z n β 1 nb 1 Γ βp nb p Γn 1 2 [ αj q Ψ 1 p,a j 1,q ; ] β j,b j 1,p ; z. 5.3 For k order we have F k [ qψ p αj,a j 1,q ; [ β j,b j 1,p ; z] Γα1 na 1 Γ ] k α q na q Γ F z n β 1 nb 1 Γ βp nb p Γn 1 k n [ Γα1 na 1 Γ ] k α q na q n Γ z n β 1 nb 1 Γ βp nb p Γn 1 k1 [ αj q Ψ k p,a j 1,q ; ] β j,b j 1,p ; z, 5.4 the Laguerre-type Fox-Wright function.

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