Azerbaijan Journal of Mathematics

Abstract. In this work, we present Chlodowsky variation of Szasz-Mirakyan-Stancu operators via (p; q) calculus and Szasz-Mirakyan-Baskakov-Stancu type operator. Here, we have calculated the moments and then formulated few properties which involves weighted approximation and direct estimates. For the particular case α = 0; β = 0; bn = 1; the previously known results for two parameter quantum-Szasz-Mirakyan operators are obtained.
KeyWords and Phrases: Szasz-Mirakyan operators, Approximation theory, Linear positive operators, (p; q) calculus.
2000 Mathematics Subject Classiffcations: 41A25, 41A35, 41A36

References.

[1] T. Acar, (p; q)-generalization of Szasz-Mirakyan operators, Math. Meth. Appl. Sci.
2015, 39(10), 2685{2695 (2016).
[2] A.R. Gairola, Deepmala, L.N. Mishra, On the q􀀀derivatives of a certain linear pos-
itive operators, Iranian Journal of Science & Technology, Transactions A: Science,
42(3), (2018), pp. 1409{1417. DOI 10.1007/s40995-017-0227-8.
[3] A.R. Gairola, Deepmala, L.N. Mishra, Rate of Approximation by Finite Iterates of q-
Durrmeyer Operators, Proceedings of the National Academy of Sciences, India Section
A: Physical Sciences, 86(2), 229{234 (2016).
[4] A. Kumar, Vandana, Some approximation properties of generalized integral type op-
erators, Tbilisi Mathematical Journal, 11(1) (2018), pp. 99116.
[5] V.N. Mishra, K. Khatri, L.N. Mishra, Deepmala, Inverse result in simultaneous ap-
proximation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and
Applications, 2013 (1), 586 (2013).
[6] V.N. Mishra, S. Pandey, On (p; q) Baskakov Durrmeyer Stancu Operators, Advances
in Applied Cliord Algebras, 27(2), 1633{1646 (2017).
[7] V.N. Mishra, K. Khatri, L.N. Mishra, On Simultaneous Approximation for Baskakov-
Durrmeyer-Stancu type operators, Journal of Ultra Scientist of Physical Sciences,
24(3) A, 2012, pp. 567-577.

[8] V.N. Mishra, K. Khatri, L.N. Mishra, Statistical approximation by Kantorovich type
Discrete q􀀀Beta operators, Advances in Dierence Equations 2013, 2013:345, DOI:
10.1186/10.1186/1687-1847-2013-345.
[9] U. Kadak, V.N. Mishra, S. Pandey, Chlodowsky Type Generalization of (p; q)-Szasz
Operators Involving Brenke Type Polynomials, Revista de la Real Academia de Ciencias
Exactas, Fsicas y Naturales. Serie A. Matemticas (RACSAM), 112(4), (2018),
pp. 1443-1462. DOI:10.1007/s13398-017-0439-y
[10] V.N. Mishra, S. Pandey, On Chlodowsky variant of (p; q) Kantorovich-Stancu-Schurer
operators, International Journal of Analysis and Applications, 11(1), pp. 28-39
(2016).
[11] L.N. Mishra, S. Pandey, V.N. Mishra, On a class of Generalised (p, q) Bernstein
operators, Indian Journal of industrial and applied mathematics , 10 (1), pp. 190-
203 (2019).
[12] A.D. Gadzhiev, Theorems of the type of P. P. Korovkin type theorems, Math. Zametki,
20(5), 1976, 781-786; English Translation, Math. Notes, 20(5/6), 996{998 (1976).
[13] R.B. Gandhi, Deepmala, V.N. Mishra, Local and global results for modied Szasz -
Mirakjan operators, Math. Method. Appl. Sci., 40(7), (2017), pp. 2491-2504. DOI:
10.1002/mma.4171.
[14] M.N. Hounkonnou, J. Desire, B. Kyemba, R(p, q)-calculus: dierentiation and inte-
gration, SUT Journal of Mathematics, 49(2), 145{167 (2013).
[15] G. Ic,oz, R. N. Mohapatra, Approximation properties by q-Durrmeyer-Stancu opera-
tors, Anal. Theory Appl., 29(4), 373{383 (2013) .
[16] R. Jagannathan, K.S. Rao, Two-parameter quantum algebras, twin-basic numbers,
and associated generalized hypergeometric series, Proceedings of the International
Conference on Number Theory and Mathematical Physics, 20-21 December 2005.
[17] R.N. Mohapatra, Z. Walczak, Remarks on a class of Szasz-Mirakyan type operators,
East J. Approx., 15(2), 197{206 (2009).
[18] M. Mursaleen, K.J. Ansari, A. Khan, On (p; q)-analogue of Bernstein operators, Appl.
Math. Comput., 266, 874{882 (2015). [Erratum: Appl. Math. Comput., 278, 70{71
(2016) ].
[19] M. Mursaleen, A. Alotaibi, K.J. Ansari, On a Kantorovich variant of (p; q)-Szasz-
Mirakjan operators, J. Function Spaces, 2016, Accession no 112724014, 9 pages.
[20] M. Mursaleen, A. Naaz, A. Khan, Improved approximation and error estimations by
King type (p; q)-Szasz-Mirakjan Kantorovich operators, Appl. Math. & Comp., 348,
175-185 (2019).

[21] M. Mursaleen, M. Nasiuzzaman, N. Ashirbayev, A. Abzhapbarov, Higher order gen-
eralization of Bernstein type operators dened by (p; q)-integers, J. Comput. Anal.
Appl., 25(5), 817{829 (2018).