Research Article Translation, Creation and Annihilation of Poles …downloads.hindawi.com/archive/2014/716718.pdf · 2019-07-31 · Research Article Translation, Creation and Annihilation

Research ArticleTranslation, Creation and Annihilation of Poles andZeros with the Biernacki and Ruscheweyh Operators, Acting onMeijer’s 𝐺-Functions

Amir Pishkoo1,2 and Maslina Darus2

1 Nuclear Science Research School, NSTRI, P.O. Box, Tehran 14395-836, Iran2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi,Selangor, Malaysia

Correspondence should be addressed to Amir Pishkoo; [email protected]

Received 17 November 2013; Accepted 2 January 2014; Published 12 February 2014

Academic Editors: S. Deng and N. Igbida

Copyright © 2014 A. Pishkoo and M. Darus. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Meijer’sG-functions are studied by the Biernacki and Ruscheweyh operators.These operators are special cases of the Erdelyi-Koberoperators (for 𝑚 = 1). The effect of operators on Meijer’s G-functions can be shown as the change in the distribution of poles andzeros on the complex plane. These poles and zeros belong to the integrand, a ratio of gamma functions, defining the Meijer’s G-function. Displacement in position and increasing or decreasing in number of poles and zeroes are expressed by the transporter,creator, and annihilator operators. With special glance, three basic univalent Meijer’sG-functions, Koebe, and convex functions areconsidered.

1. Introduction

In studying of analytic functions, consideration of existingzeros and poles on the complex plane is the first basic step.Through all analytic functions, Gamma function Γ(𝑧) hasinfinity poles at 𝑛 = 0, −1, −2, and so forth, whereas it doesnot have any zeros [1].

Consider the following:

Γ (𝑧) = lim𝑛→∞

𝑛!𝑛𝑧

𝑧 (𝑧 + 1) ⋅ ⋅ ⋅ (𝑧 + 𝑛)

. (1)

It is meromorphic function, meaning that it is analyticexcept for isolated singularities which are poles. However, thefunction 1/Γ(𝑧) does not have any poles, instead it has infinityzeroes.This function is an entire function, and itsWeierstrassproduct is [1]

1

Γ (𝑧)

= 𝑧𝑒𝛾𝑧

∞

∏

𝑛=1

(1 +

𝑧

𝑛

) 𝑒−𝑧/𝑛

, (2)

where 𝛾 is known as the Euler constant 𝛾. Gamma functionwith different argument has infinity poles in other places. In

integral definition of Meijer’s 𝐺-functions, we face the prod-ucts of Gamma functions in numerator and denominator.

Meijer’s 𝐺-function has been useful in mathematicalphysics because of its analytical properties and because itcan be expressed as a finite number of generalized hyperge-ometric functions which have well-known series expansions.These functions are defined as follows.

Definition 1. A definition of the Meijer’s 𝐺-function is givenby the following path integral in the complex plane [2–6]:

𝐺𝑚,𝑛

𝑝,𝑞

𝑎1, . . . , 𝑎

𝑝

𝑏1, . . . , 𝑏

𝑞

| 𝑧

=

1

2𝜋𝑖

∫

𝐿

∏𝑚

𝑗=1Γ (𝑏𝑗− 𝑠) × ∏

𝑛

𝑗=1Γ (1 − 𝑎

𝑗+ 𝑠)

∏𝑞

𝑗=𝑚+1Γ (1 − 𝑏

𝑗+ 𝑠) × ∏

𝑝

𝑗=𝑛+1Γ (𝑎𝑗− 𝑠)

𝑧𝑠

𝑑𝑠.

(3)

This integral is included in the so-called Mellin-Barnes typeand may be viewed as an inverse Mellin transform. Here, anempty productmeans unity, and the integers𝑚, 𝑛, 𝑝, and 𝑞 are

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2014, Article ID 716718, 5 pageshttp://dx.doi.org/10.1155/2014/716718

2 Chinese Journal of Mathematics

called the “orders” of the𝐺-function or the components of theorder (𝑚; 𝑛; 𝑝; 𝑞). Here, 𝑎

𝑝and 𝑏𝑞are called “parameters” and,

generally, they are complex numbers. The definition holdsunder the following assumptions: 0 ≤ 𝑚 ≤ 𝑞 and 0 ≤

𝑛 ≤ 𝑝, where 𝑚, 𝑛, 𝑝, and 𝑞 are integer numbers. Further,𝑎𝑗− 𝑏𝑘= 1, 2, 3, . . . for 𝑘 = 1, . . . , 𝑛 and 𝑗 = 1, 2, . . . , 𝑚 imply

that no pole of any Γ(𝑏𝑗− 𝑠), 𝑗 = 1, . . . , 𝑚 coincides with any

pole of any Γ(1 − 𝑎𝑘+ 𝑠), 𝑘 = 1, . . . , 𝑛.

Based on the definition, the following basic property iseasily derived:

𝑧𝛼

𝐺𝑚,𝑛

𝑝,𝑞

apbq

| 𝑧𝐺𝑚,𝑛

𝑝,𝑞

ap + 𝛼bq + 𝛼

| 𝑧, (4)

where the multiplying term 𝑧𝛼 changes the parameters of the

𝐺-function.

Definition 2 (see [1, 7]). TheMellin transform of a function isgiven by

𝐹 (𝑠) = ∫

∞

0

𝑥𝑠−1

𝑓 (𝑥) 𝑑𝑥. (5)

Definition 3 (see [1, 7]). The inverseMellin transform is givenby

𝑓 (𝑥) =

1

2𝜋𝑖

∫

𝑐+𝑖∞

𝑐−𝑖∞

𝑥−𝑠

𝐹 (𝑠) 𝑑𝑠, 𝑐 > 0. (6)

In [8] Pishkoo and Darus assign a more substantive roleto Meijer’s 𝐺-functions in the univalent functions theory.They use some selected Meijer’s 𝐺-functions as univalentMeijer’s 𝐺-functions and path integral representation insteadof their representative series form. The results are shown intables wherein each univalentMeijer’s𝐺-function of an upperrank is obtained from another univalent Meijer’s 𝐺-functionof lower rank. Herein, and overall, it so suffices that weconsider the three basic univalent Meijer’s 𝐺-functions: 𝐺1,0

0,2;

𝐺1,1

1,2; 𝐺1,11,1

and from which a number of univalent Meijer’s 𝐺-functions can be obtained. One of the most important resultsin the work mentioned above is that the Erdelyi-Kober (E-K)operators (for𝑚 = 1, 2) relate Meijer’s 𝐺-functions together.

In [9], Kiryakova et al. applied generalized fractionalcalculus in univalent function theory and derived equiva-lent forms of some well-known operators in terms of E-Kfractional integral and derivative.The Biernacki operator wasobtained in terms of the E-K fractional integral.


𝐵𝑓 (𝑧) = ∫

𝑧

0

𝑓 (𝜉)

𝜉

𝑑𝜉 = ∫

1

0

𝑓 (𝑧𝜎)

𝜎

𝑑𝜎 = 𝐼−1,1

1𝑓 (𝑧) . (7)

Meanwhile, the so-called Ruscheweyh derivative defined by

𝐷𝛼

𝑓 (𝑧) = (

𝑧

(1 − 𝑧)1+𝛼

∗ 𝑓) (𝑧) (𝛼 ≥ 0) (8)

was also obtained in terms of the E-K fractional derivative oforder 𝛼, as follows:

𝐷𝛼

𝑓 (𝑧) =

1

Γ (𝛼 + 1)

[𝑧(

𝑑

𝑑𝑧

)

𝛼

𝑧𝛼−1

]𝑓 (𝑧)

=

1

Γ (𝛼 + 1)

𝐷−1,𝛼

1𝑓 (𝑧) .

(9)

If 𝛼 = 1, then we have

𝐷1

𝑓 (𝑧) = 𝑧

𝑑

𝑑𝑧

𝑓 (𝑧) = 𝐷−1,1

1𝑓 (𝑧) . (10)

referring to (3), path integral definition of Meijer’s 𝐺-function, for each 𝐺-function there exists special integrandthat has poles and zeroes which can be shown by theirpositions. In [10, 11] we show that Meijer’s 𝐺-functions arethe solution of physical models and in micro- and nanos-tructures. In quantum mechanics, operators are measurablephysical quantities and wave functions as physical systems.In this paper we study Meijer’s 𝐺-function as appropriatecandidate for describing physical systems and operatorsprepared in the language of “Generalized fractional calculusoperators” which are observables. Here, studying distributionof poles and zeroes before and after action of the operator onMeijer’s 𝐺-function leads to three definitions for operators;namely, transporter, creator, and annihilator operators.

2. Preliminaries

Using (3) for three basic univalent 𝐺-functions 𝐺1,00,2; 𝐺1,11,2;

𝐺1,1

1,1, the following are obtained.

2.1. The First Basic Univalent 𝐺-Function. Consider the fol-lowing:

𝐺1,0

0,2[

−

𝑏1, 𝑏2

| 𝑧] =

1

2𝜋𝑖

∫

𝐿

Γ (𝑏1− 𝑠) 𝑧𝑠

Γ (1 − 𝑏2+ 𝑠)

𝑑𝑠. (11)

Position of poles: 𝑠 = 𝑏1+ 𝑛; 𝑛 = 0, 1, 2, . . .

Position of zeroes: 𝑠 = 𝑏2− 1 − 𝑛; 𝑛 = 0, 1, 2, . . .

If 𝑏1= 0, 𝑏2= 1/2, and 𝑧 → 𝑧

2

/4, then then (see Figure 1)we get

cos 𝑧 = √𝜋( 1

2𝜋𝑖

)∫

𝐿

Γ (−𝑠) 𝑧2𝑠

4𝑠Γ (1/2 + 𝑠)

𝑑𝑠. (12)

If 𝑏1= 1/2, 𝑏

2= 0, and 𝑧 → 𝑧

2

/4, then (see Figure 2) weget

sin 𝑧 = √𝜋( 1

2𝜋𝑖

)∫

𝐿

Γ (1/2 − 𝑠) 𝑧2𝑠

4𝑠Γ (1 + 𝑠)

𝑑𝑠. (13)

2.2. The Second Basic Univalent 𝐺-Function. Consider thefollowing:

𝐺1,1

1,2[

𝑎1

𝑏1, 𝑏2

| 𝑧] =

1

2𝜋𝑖

∫

𝐿

Γ (𝑏1− 𝑠) Γ (1 − 𝑎

1+ 𝑠) 𝑧𝑠

Γ (1 − 𝑏2+ 𝑠)

𝑑𝑠.

(14)

Chinese Journal of Mathematics 3

0

0.5

1

0 1 2 3 4 5−5 −4 −3 −2 −1

−1

−0.5

Poles of Γ(−s)Zeroes of 1/Γ(1/2 + s)

Figure 1: Poles and zeroes related to cos 𝑧.

0

0.5

1

0 1 2 3 4 5−5 −4 −3 −2 −1

−1

−0.5

Poles of Γ(1/2 − s)Zeroes of 1/Γ(1 + s)

Figure 2: Poles and zeroes related to sin 𝑧.

Position of poles: 𝑠 = 𝑏1+𝑛; 𝑛 = 0, 1, 2, . . . and 𝑠 = 𝑎

1−1−𝑛;

𝑛 = 0, 1, 2, . . .

Position of zeroes: 𝑠 = 𝑏2− 1 − 𝑛; 𝑛 = 0, 1, 2, . . .

If 𝑎1= 1 + 𝑖, 𝑏

1= 0, and 𝑏

2= 𝑖 + 1/2 then (see Figure 3)

we get

𝐺1,1

1,2[

1 + 𝑖

0, 𝑖 +

1

2

| 𝑧] =

1

2𝜋𝑖

∫

𝐿

Γ (−𝑠) Γ (−𝑖 + 𝑠)

Γ (1/2 + 𝑖 + 𝑠)

𝑧𝑠

𝑑𝑠. (15)

If 𝑎1

= 𝑏2then 𝐺

1,1

1,2[

𝑎1

𝑏1,𝑏2| 𝑧] = 𝐺

1,0

0,1[−

𝑏1| 𝑧] =

(1/2𝜋𝑖) ∫𝐿

Γ(𝑏1− 𝑠)𝑧𝑠

𝑑𝑠.Position of poles: 𝑠 = 𝑏

1+ 𝑛; 𝑛 = 0, 1, 2, . . .

Position of zeroes: there are no zeroes.If we put 𝑏

1= 0, then we get exponential function

𝑒−𝑧

= 𝐺1,0

0,1[0 | 𝑧] =

1

2𝜋𝑖

∫

𝐿

Γ (−𝑠) 𝑧𝑠

𝑑𝑠. (16)

2.3. The Third Basic Univalent 𝐺-Function. Consider thefollowing:

𝐺1,1

1,1[

𝑎1

𝑏1

| 𝑧] =

1

2𝜋𝑖

∫

𝐿

Γ (𝑏1− 𝑠) Γ (1 − 𝑎

1+ 𝑠) 𝑧𝑠

𝑑𝑠. (17)

0

0.5

1

0 1 2 3 4−4 −3 −2 −1

−1

−0.5

Poles of Γ(−s)Poles of Γ(−i + s)Zeroes of 1/Γ(1/2 + i + s)

Figure 3: Poles and zeroes related to 𝐺1,11,2[1+𝑖

0,𝑖+1/2| 𝑧].

Position of poles: 𝑠 = 𝑏1+𝑛; 𝑛 = 0, 1, 2, . . . and 𝑠 = 𝑎

1−1−𝑛;

𝑛 = 0, 1, 2, . . .

Position of zeroes: there are no zeroes.In [2] all elementary functions can be expressed in terms

of 𝐺-functions. For instance

(1 − 𝑧)−𝛼

=

1

Γ (𝛼)

𝐺1,1

1,1[

1 − 𝛼

0| 𝑧] , (18)

wherein, if 𝛼 = 2, then (18) implies that

1

(1 − 𝑧)2= 𝐺1,1

1,1[

−1

0| 𝑧] . (19)

It is well known that the Koebe function plays an importantrole in the theory of univalent functions. Using (17), (19), and(4), if we put 𝑎

1= 0 and 𝑏

1= 1 then the Koebe function can

be obtained (see Figure 4) as follows:

𝐾 (𝑧) =

𝑧

(1 − 𝑧)2= 𝐺1,1

1,1[

0

1| 𝑧]

=

1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (1 + 𝑠) 𝑧𝑠

𝑑𝑠.

(20)

The path of integration is curved to separate the poles of Γ(1−𝑠) from the poles of Γ(1+𝑠). Using (17), (18), and (4), if we put𝛼 = 1, 𝑎

1= 1, and 𝑏

1= 1 then the function 𝑧/(1 − 𝑧) can be

obtained as follows:

𝑧

1 − 𝑧

= 𝐺1,1

1,1[

1

1| 𝑧] =

1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (𝑠) 𝑧𝑠

𝑑𝑠. (21)

The path of integration is curved to separate the poles of Γ(1−𝑠) from the poles of Γ(𝑠).

3. Results and Discussion

The E-K fractional derivative of order 1, namely, 𝐷−1,11

, maps𝑔(𝑧) = 𝑧/(1 − 𝑧) onto the Koebe function.

4 Chinese Journal of Mathematics

0

0.5

1

0 1 2 3 4 5−5 −4 −3 −2 −1

−1

−0.5

Poles of Γ(1 − s)Poles of Γ(1 + s)

Figure 4: Poles related to the Koebe function.

0

0.5

1

0 1 2 3 4 5−5 −4 −3 −2 −1

−1

−0.5

Poles of Γ(1 − s)Poles of Γ(s)

Figure 5: Poles related to the function 𝑔(𝑧).

Consider (see Figure 5) the following:

𝑔 (𝑧) =

𝑧

1 − 𝑧

= 𝐺1,1

1,1

1

1| 𝑧 =

1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (𝑠) 𝑧𝑠

𝑑𝑠.

1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (1 + 𝑠) 𝑧𝑠

𝑑𝑠

= 𝐷−1,1

1(

1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (𝑠) 𝑧𝑠

𝑑𝑠) .

(22)

So we obtain

𝐷−1,1

1𝐺1,1

1,1

1

1| 𝑧 ≻ 𝐺

1,1

1,1[

1

1| 𝑧] (23)

or

𝐺1,1

1,1

1

1| 𝑧 ≺ 𝐺

1,1

1,1[

0

1| 𝑧] (24)

Vice versa, the E-K fractional integral, the Biernackioperator 𝐼−1,1

1, maps theKoebe function onto𝑔(𝑧) = 𝑧/(1−𝑧).


1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (𝑠) 𝑧𝑠

𝑑𝑠

= 𝐼−1,1

1(

1

2𝜋𝑖

∫

𝐿

Γ (1 − 𝑠) Γ (1 + 𝑠) 𝑧𝑠

𝑑𝑠) .

(25)

Similarly we have

𝐼−1,1

1𝐺1,1

1,1

0

1| 𝑧 ≺ 𝐺

1,1

1,1[

0

1| 𝑧] . (26)

And more than this we have

𝐼−1,1

1𝐷−1,1

1𝐺1,1

1,1

1

1| 𝑧 = 𝐺

1,1

1,1[

1

1| 𝑧] ,

𝐷−1,1

1𝐼−1,1

1𝐺1,1

1,1

0

1| 𝑧 = 𝐺

1,1

1,1[

0

1| 𝑧] .

(27)

Definition 4. Pole (zero) transporter is the Erdelyi-Koberoperator that changes the argument of Gamma function(s)which is (are) inside the contour integral definition ofMeijer’s𝐺-function and shifts position of poles (zeros) of Gammafunction(s) on the complex plane.

Definition 5. Pole (zero) creator is the Erdelyi-Kober operatorthat creates excess Gamma function(s), on the numerator(denominator) of the integrand, inside the contour integraldefinition of Meijer’s 𝐺-function and can create a few orinfinity excess poles (zeroes) on the complex plane.

Definition 6. Pole (zero) annihilator is the Erdelyi-Koberoperator that annihilates Gamma function(s), on the numer-ator (denominator) of the integrand, inside the contourintegral definition of Meijer’s 𝐺-function and can annihilatesa few or infinity poles (zeroes) on the complex plane.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported by MOHE with the Grant no.ERGS/1/2013/STG06/UKM/01/2.

References

[1] G. Arfken and H. Weber, Mathematical Methods for Physicists,Elsevier, Oxford, UK, 2001.

[2] V. Kiryakova, Generalized Fractional Calculus and Applications,Longman, Harlow, UK, 1994.

[3] L. C. Andrews, Special Functions for Engineers and AppliedMathematicians, MacMillan, New York, NY, USA, 1985.

[4] Y. L. Luke,The Special Functions andTheir Approximations, vol.1 of Complex Variables, Academic Press, New York, NY, USA,1969.

Chinese Journal of Mathematics 5

[5] A. U. Klimyik, “Meijer G-function,” in Encyclopaedia of Mathe-matics, Springer, Berlin, Germany, 2010.

[6] R. A. Askey, “Meijer G-function,” in NIST Handbook of Mathe-matical Functions, Cambridge University Press, New York, NY,USA, 2010.

[7] S. Lang, Complex Analysis, Springer, Berlin, Germany, 1985.[8] A. Pishkoo andM.Darus, “Fractional differintegral transforma-

tion of univalent Meijer’s G-functions,” Journal of Inequalitiesand Applications, vol. 2012, article 36, 2012.

[9] V. Kiryakova, M. Saigo, and H. M. Srivastava, “Some crite-ria for univalence of analytic functions involving generalizedfractional calculus operators,” Fractional Calculus and AppliedAnalysis, vol. 1, pp. 79–104, 1998.

[10] A. Pishkoo and M. Darus, “Meijer’s G-functions (MGFs) inMicro- and Nano-structures,” Journal of Computational andTheoretical Nanoscience, vol. 10, no. 10, pp. 2478–2483, 2013.

[11] A. Pishkoo and M. Darus, “Some applications of Meijer G-functions as solutions of differential equations in physicalmodels,” Journal of Mathematical Physcis, Analysis, Geometry,vol. 3, no. 9, pp. 379–391, 2013.

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Research Article Translation, Creation and Annihilation of Poles …downloads.hindawi.com/archive/2014/716718.pdf · 2019-07-31 · Research Article Translation, Creation and Annihilation