Quasi-monomiality and convergence theorem for the Boas-Buck-Sheffer polynomials

Shahid Ahmad Wani; Kottakkaran Sooppy Nisar; Shahid Ahmad Wani; Kottakkaran Sooppy Nisar

doi:10.3934/math.2020283

AIMS Mathematics

2020, Volume 5, Issue 5: 4432-4443. doi: 10.3934/math.2020283

Previous Article Next Article

Research article

Quasi-monomiality and convergence theorem for the Boas-Buck-Sheffer polynomials

Shahid Ahmad Wani ¹,
Kottakkaran Sooppy Nisar ^{2
,
,}

1.
Department of Mathematics, Aligarh Muslim University, Aligarh, India
2.
Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia

Received: 22 December 2019 Accepted: 07 May 2020 Published: 12 May 2020
MSC : 33C45, 33C99, 33E20

A mixed family of polynomials, called the Boas-Buck-Sheffer family is introduced and their quasi-monomial properties are established in this article. Also, the generalizations of the Szasz operators including this mixed polynomial family are obtained and their convergence is studied.
- Boas-Buck polynomials,
- monomiality principle,
- Szasz operators
Citation: Shahid Ahmad Wani, Kottakkaran Sooppy Nisar. Quasi-monomiality and convergence theorem for the Boas-Buck-Sheffer polynomials[J]. AIMS Mathematics, 2020, 5(5): 4432-4443. doi: 10.3934/math.2020283

Related Papers:

Abstract

A mixed family of polynomials, called the Boas-Buck-Sheffer family is introduced and their quasi-monomial properties are established in this article. Also, the generalizations of the Szasz operators including this mixed polynomial family are obtained and their convergence is studied.

References

[1]	I. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J., 5 (1939), 590-622. doi: 10.1215/S0012-7094-39-00549-1
[2]	A. Di Bucchianico, D. Loeb, A selected survey of umbral calculus, Electron. J. Combin, 2 (1995) 28.
[3]	T. X. He, L. C. Hsu, P. J. S. Shiue, The Sheffer group and the Riordan group, Discrete Appl. Math., 155 (2007), 1895-1909. doi: 10.1016/j.dam.2007.04.006
[4]	W. Wang, A determinantal approach to Sheffer sequences, Linear Algebra Appl., 463 (2014), 228-254. doi: 10.1016/j.laa.2014.09.009
[5]	S. Khan, M. W. Al-Saad, G. Yasmin, Some properties of Hermite-based Sheffer polynomials, Appl. Math. Comput., 217 (2010), 2169-2183.
[6]	S. Khan, N. Raza, Monomiality principle, operational methods and family of Laguerre-Sheffer polynomials, J. Math. Anal. Appl., 387 (2012), 90-102.
[7]	S. Khan, N. Raza, Families of Legendre-Sheffer polynomials, J. Math. Comput. Modell., 55 (2012), 969-982. doi: 10.1016/j.mcm.2011.09.023
[8]	S. Khan, M. Riyasat, Determinantal approach to certain mixed special polynomials related to Gould-Hopper polynomials, Appl. Math. Comput., 251 (2015), 599-614.
[9]	S. Khan, G. Yasmin, N. Ahmad, On a new family related to truncated exponential and Sheffer polynomials, J. Math. Anal. Appl., 418 (2014), 921-937. doi: 10.1016/j.jmaa.2014.04.028
[10]	S. Roman, The umbral calculus, Academic Press, New York, 1984.
[11]	R. P. Boas, R. C. Buck, Polynomials defined by generating relations, Am. Math. Mon., 63 (1956), 626-632. doi: 10.1080/00029890.1956.11988880
[12]	W. C. Brenke, On generating functions of polynomial systems, Am. Math. Mon., 52 (1945), 297-301. doi: 10.1080/00029890.1945.11991572
[13]	E. D. Rainville, Special functions, Springer Tracts in Modern Physics, New York, 1971.
[14]	P. Appell, Sur une classe de polynômes, Annales Scientifiques de l'École Normale Supérieure, 1880.
[15]	J. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math., 73 (1941), 333-366. doi: 10.1007/BF02392231
[16]	G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle, Adv. Spec. Funct. Appl., (1999), 147-164.
[17]	Y. B. Cheikh, Some results on quasi-monomiality, Appl. Math. Comput., 141 (2003), 63-76.
[18]	O. Szasz, Generalization of S. Bernstein's polynomials to the infinite interval, J. Res. Nat. Bur. Standards., 45 (1950), 239-245. doi: 10.6028/jres.045.024
[19]	A. Jakimovski, D. Leviatan, Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 34 (1969), 97-103.
[20]	M. Ismail, On a generalization of Szász operators, Mathematica (Cluj), 39 (1974), 259-267.
[21]	S. Sucu, G. İçöz, S. Varma, On some extensions of Szász operators including Boas-Buck-type polynomials, Abstr. Appl. Anal., 2012 (2012), 1-15.
[22]	S. Khan, M. Riyasat, 2-iterated Sheffer polynomials, arXiv: Classical Anal. ODEs, 2015.
[23]	P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl Akad Nauk SSSR, 90 (1996), 961-964.
[24]	I. Gavrea, I. Raşa, Remarks on some quantitative Korovkin-type results, Rev. Anal. Numér. Théor. Approx., 28 (1993) 173-176.
[25]	V. Zhuk, Functions of the Lip1 class and SN Bernstein's polynomials, Vestnik Leningrad Univ. Mat. Mekh. Astronom., 1 (1989), 25-30.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)