On a New Approach by Modified (p; q)-Szasz-Mirakyan Operators

Main Article Content

Vishnu Narayan Mishra, Ankita R. Devdhara, Khursheed J. Ansari, Seda Karateke

Abstract

In this paper, we introduce a new type of (p; q) exponential function with some properties and a modified (p; q)-Szasz-Mirakyan operators by virtue of this function by investigating approximation properties. We obtain moments of generalized (p; q)-Szasz-Mirakyan operators. Furthermore, we derive direct results, rate of convergence, weighted approximation result, statistical convergence and Voronovskaya type result of these operators with numerical examples. Graphical representations reveal that modified (p; q)-Szasz-Mirakyan operators have a better approximation to continuous functions than pioneer one.

Article Details

References

  1. J.L. Cieslinski, Improved q-exponential and q-trigonometric functions, Appl. Math. Lett. 24 (2011), 2110-2114.
  2. M. Mursaleen, K.J. Ansari and A. Khan, On (p; q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2015), 874-882. [Erratum: Appl. Math. Comput. 278 (2016) 70-71].
  3. M. Mursaleen, K.J. Ansari and A. Khan, Some approximation results by (p; q)-analogue of Bernstein-Stancu operators, Appl. Math. Comput. 264 (2015), 392-402. [Corrigendum: Appl. Math. Comput 269 (2015) 744-746].
  4. M. Mursaleen, A. Alotaibi and K.J. Ansari, On a Kantorovich Variant of (p; q)-Szasz-Mirakjan Operators, J. Funct. Spaces, 2016 (2016), 1035253.
  5. T. Acar, (p; q)-generalization of Szasz-Mirakyan operators, Math. Meth. Appl. Sci. 39 (10) (2016), 2685-2695.
  6. R. Jagannathan, K.S. Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, Proc. Int. Conf. Numb. Theory Math. Phys. Kumbakonam, India, 2005.
  7. R.B. Corcino, On p,q-binomial coefficients, Integers: Electron. J. Comb. Numb. Theory. 8 (2008), #A29.
  8. V. Gupta, A. Aral, Convergence of the q-analogue of Szasz-Beta operators. Appl. Math. Comput. 216 (2010), 374-380.
  9. R.A. Devore, G.G. Lorent, Constructive approximation, Springer, Berlin, 1993.
  10. A.D. Gadjev, C. Orham, Some approximation theorems via statistical convergence, Rocky Mt. J. Math. 32(1) (2002) 129-138.
  11. R.B. Gandhi, Deepmala, V.N. Mishra, Local and global results for modified Szasz-Mirakjan operators, Math. Method. Appl. Sci., 40(7) (2017), 2491-2504.
  12. M. Mursaleen, AAH. Al-Abied and A. Alotaibi, On (p; q) Szasz-Mirakyan operators and their approximation properties. J. Inequal. Appl. 2017 (2017), 196.