On the representation of analytic functions by series of derived bases of polynomials in hyperelliptical regions

Al-Sheikh, Mohamed S.; Hassan, Gamal F.; Ibrahim, Abd Almonem M.; Zahran, Ahmed M.

doi:10.21608/absb.2022.111238.1162

On the representation of analytic functions by series of derived bases of polynomials in hyperelliptical regions

Document Type : Original Article

Authors

¹ Mathematics department, Faculty of Science, Al-Azhar University, Assiut branch, Egypt.

² Mathematics department, Faculty of Science, Assiut University, Assiut, Egypt.

10.21608/absb.2022.111238.1162

Abstract

One of the important themes in complex analysis is the expansion of analytic functions by infinite series in a given sequence of bases of polynomials. In the present paper, we investigated the representation of analytic functions in different domains of derived bases of polynomials. The behaviour of the associated representation of whole functions is directly related to determining the convergence properties (effectiveness) of such bases. The representation domains are closed hyperellipses, open hyperellipses, and closed regions surrounding a closed hyperellipse. Also, some results concerning the order of derived bases in hyperellipse are obtained. The results obtained are natural generalisations of the results obtained in hyperspherical regions.

Keywords

Main Subjects

Mathematics

References

[1] Whittaker JM. On series of polynomials. The Quarterly Journal of Mathematics. 1934 01;os- 5(1):224-39. Available from: https://doi.org/10.1093/qmath/os-5.1.224.

[2] Abul-Ez MA. Bessel Polynomial Expansions in Spaces of Holomorphic Functions. Journal of Math- ematical Analysis and Applications. 1998;221(1):177-90. Available from: https://www.sciencedirect. com/science/article/pii/S0022247X97958406.

[3] Aloui L, Abul-Ez MA, Hassan GF. Bernoulli special monogenic polynomials with the difference and sum polynomial bases. Complex Variables and Elliptic Equations. 2014;59(5):631-50. Available from: https://doi.org/10.1080/17476933.2012.750450.

[4] Aloui L, Abul-Ez MA, Hassan GF. On the order of the difference and sum bases of polynomials in Clifford setting. Complex Variables and Elliptic Equations. 2010;55(12):1117-30. Available from: https://doi.org/10.1080/17476931003728339.

[5] Hassan GF, Aloui L. Bernoulli and Euler Polynomials in Clifford Analysis. Advances in Applied Clifford Algebras.2015;25:351-76.

[6] Abul-Dahab M. On the Construction of Generalized Monogenic Bessel Polynomials.

2018 07.

[7]Abul-Ez MA, Zayed M ''Criteria in Nuclear Fr'echet spaces and Silva spaces with refinement of the Cannon–Whittaker theory.'' J. Funct. Spaces, {2020}, 15 (2020).

[8] Cannon B. On the convergence of series of polynomials. Journal of the London MathematicalSociety.1939;s1-14(1):51-62.Availablefrom:https://londmathsoc.onlinelibrar y.wiley.com/doi/abs/10.1112/ jlms/s1-14.1.51.

[9] Cannon B. On the convergence of integral functions by general basic series. Mathematisch Zeitschrift. 1939;45:158-205.

[10] Mursi M, Maker RH. Basic sets of polynomials of several complex variables. In The Second Arab Science. 1955;2:51-68.

[11] Nassif M. Composite sets of polynomials of several complex variables. Publications Mathematics. 1971;18:43-52.

[12] Abul-Ez M, Abd-Elmageed H, Hidan M, Abdalla M. On the Growth Order and Growth Type of Entire Functions of Several Complex Matrices. Journal of Function Spaces. 2020;vol. 2020:9.

[13] Abul-Ez M, Zayed M. Criteria in Nuclear Frchet Spaces and Silva Spaces with Refinement of the Cannon-Whittaker Theory. Journal of Function Spaces. 2020;vol. 2020:15.

[14] Sayed SA. Hadamard Product of Simple Sets of Polynomials in Cn. Theory and Applications of Mathematics & Computer Science. 2014;4:26-39.

[15] Kishka ZMG, EL-sayed A. On the effectiveness of basic sets of polynomials of several complex variables in elliptical regions. Proceedings of the 3rd International ISAAC Congress. 2003:265-78.

[16] Hassan GF. Ruscheweyh differential operator sets of basic sets of polynomials of several complex variables in hyperelliptical regions. Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis. 2006;20:247-64.

[17] Kishka ZMG, Saleem M, Abul-Dahab M. On Simple Exponential Sets of Polynomials. Mediterranean Journal of Mathematics. 2014 05;11.

[18] Kumuyi WF, Nassif M. Derived and integrated sets of simple sets of polynomials in two complex variables. Journal of Approximation Theory. 1986;47:270-83.

[19] Ahmed AeS. On some classes and spaces of holomorphic and hyperholomorphic functions.

Weimar, Bauhausuniv, Diss. 2003.

[20] Abul-ez MA, Constales D. Basic sets of pofynomials in clifford analysis. Complex Variables, Theory and Application: An International Journal. 1990;14(1-4):177-85. Available from: https://doi.org/10. 1080/17476939008814416.

[21] Abul-Ez MA. Exponential base of special monogenic polynomials. Pure Mathematics And Applications.1997;8(2-4):137-46.Availablefrom:https://ideas.repec.org/a/cmt/pumat/ puma1997v008pp0137-0146.html.

[22] Abul-Ez MA. Hadamard product of bases of polynomials in clifford analysis. Complex Variables, Theory and Application: An International Journal. 2000;43(2):109-28. Available from: https://doi. org/10.1080/17476930008815304.

[23] Abul-Ez MA, Constales D. Similar functions and similar bases of polynomials in clifford setting. Complex Variables, Theory and Application: An International Journal. 2003;48(12):1055-70. Avail- able from:https://doi.org/10.1080/02781070310001634584.

[24] Abul-Ez MA, Constales D. The square root base of polynomials in Clifford analysis. Archiv der Mathematik. 2003;80(5):486-95.

[25] Abul-Ez M, Constales D, Morais J, Zayed M. Hadamard three-hyperballs type theorem and over- convergence of special monogenic simple series. Journal of Mathematical Analysis and Applications. 2014;412(1):426-34. Available from: http://dx.doi.org/10.1016/j.jmaa.2013.10,068.

[26]Aloui L, Hassan GF. Hypercomplex derivative bases of polynomials in Clifford analysis.MathematicalMethodsintheAppliedSciences.2010;33(3):350-7.Availablefrom:https://onlinelibrary.wiley.com/ doi/abs/10.1002/mma.1176.

[27] Saleem MA, Abul-Ez M, Zayed M. On polynomial series expansions of Cliffordian functions. Math- ematical Methods in the Applied Sciences. 2012;35(2):134-43. Available from: https://onlinelibrary. wiley.com/doi/abs/10.1002/mma.1546.

[28] Zayed M. Generalized Hadamard product bases of special monogenic polynomials. Advances in Applied Clifford Algebras. 2020;30.

[29] Zayed M. Certain Bases of Polynomials Associated with Entire Functions in Clifford Analysis. Ad- vances in Applied Clifford Algebras. 2021 04;31.

[30] Hassan GF, Aloui L, Bakali A. Basic Sets of Special Monogenic Polynomials

in Fr'echet Modules. Journal of Complex Analysis. 2017 02:1-11.

[31] Adepoju JA, Nassif M. Effectiveness of transposed inverse sets in faber regions. International Journal of Mathematics and Mathematical Sciences. 1983;6:285-95.

[32] Sayyed K. Effectiveness of similar sets of polynomials of two complex variables in polycylinders and in Faber regions. International Journal of Mathematics and Mathematical Sciences. 1998 01;21.

[33] Newns WF. On the representation of analytic functions by in_nite series.

Mathematical and Physical Sciences. 1953:429-68

[34] Makar R H. ''On derived and integral basic sets of polynomials'' Proc. Amer. Math. Soc, 5 218-225 (1954)

[35] Mikhail MN. ''Derived and integral sets ofbasic sets of polynomials'' 1953:4 :251-259.

[36] Zayed M, Abul-Ez M, Morais J. Generalized Derivative and Primitive of Cliffordian Bases of Polynomials Constructed Through Appell Monomials.Computational Methods and Function

Theory. 2012 12;12.

[37] Abul-Ez MA, Constales D. On the Order of Basic Series Representing Clifford Valued Functions. Appl Math Comput.2003oct;142(23):575584.Availablefrom: https://doi.org/10.1016/S0096-3003(02) 00350-8.

[38] Hassan G. A note on the growth order of the inverse and product bases of special monogenic polyno- mials. Mathematical Methods in the Applied Sciences. 2012 02;35:286 292.

[39] Kishka ZMG, Ahmed AES. On the order and type of basic and composite sets of polynomials in complete Reinhardt domains. Periodica Mathematica Hungarica. 2003;46:67-79.

[40] Metwally MS. Derivative operator on a hypergeometric function of three complex variables. Southwest Journal of Pure and Applied Mathematics [electronic only]. 1999;2:42-6. Available from: http://eudml. org/doc/226870