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On a \(\psi \)-Generalized Fractional Derivative Operator of Riemann–Liouville with Some Applications

Abstract

Various extended beta functions have been investigated by many researchers (Mubeen et al. in FJMS 102:1545–1557, 2017; Özergin et al. in J Comput Appl Math 235:4601–4610, 2011; Parmar et al. in J Class Anal 11:91–106, 2017). Our aim in this paper is to develop \(\psi \)-generalized Riemann Liouville fractional derivative operator by using \(\psi \)-generalized beta function. We also present some new results on \(\psi \)-generalized beta function and introduce some integral representations of \(\psi \)-generalized hypergeometric functions. Moreover, we establish fractional derivative formulas of some known functions and discuss the properties of Mellin transform. Further we discuss some applications based on generating relation for the \(\psi \)-generalized hypergeometric functions with the help of new \(\psi \)-generalized Riemann–Liouville fractional derivative operator.

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References

  1. 1.

    Agarwal, P., Choi, J., Parisc, R.B.: Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 8, 451–466 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ata, E.: Generalized Beta Function Defined by Wright function. arXiv:1803.03121v1 [math.CA] (2018)

  3. 3.

    Baleanu, D., Agarwal, P., Parmar, R.K., Alqurashi, M.M., Salahshour, S.: Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 10, 2914–2924 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bansal, M.K., Kumar, D.: On the integral operators pertaining to a family of incomplete I-functions. AIMS Math. 5(2), 1247–1259 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bansal, M.K., Kumar, D., Jain, R.: A study of Marichev Saigo Maeda fractional integral operators associated with S-generalized Gauss hypergeometric function. Kyungpook Math. J. 59, 433–443 (2019)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bansal, M.K., Kumar, D., Nisar, K.S., Singh, J.: Certain fractional calculus and integral transform results of incomplete \(\aleph \)-functions with applications. Math Methods Appl. Sci. 43, 5602–5614 (2020)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bohner, M., Rahman, G., Mubeen, S., Nisar, K.S.: A further extension of the extended Riemann–Liouville fractional derivative operator. Turk. J. Math. 42, 2631–2642 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chaudhry, M.A., Qadir, A., Rafique, M., Zubair, S.M.: Extension of Euler’s beta function. J. Comput. Appl. Math. 78, 19–32 (1997)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chaudhry, M.A., Qadir, A., Srivastava, H.M., Paris, R.B.: Extended hypergeometric and confiuent hypergeometric functions. Appl. Math. Comput. 159, 589–602 (2004)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chaudhry, M.A., Zubair, S.M.: Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55, 99–124 (1994)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  12. 12.

    Mubeen, S., Rahman, G., Nisar, K.S., Choi, J., Arshad, M.: An extended beta function and its properties. FJMS 102, 1545–1557 (2017)

    Article  Google Scholar 

  13. 13.

    Özergin, E., Özarslan, M.A., Altin, A.: Extension of gamma, beta and hypergeometric functions. J. Comput. Appl. Math 235, 4601–4610 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Özarslan, M.A., Özergin, E.: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator. Math. Comput. Model. 52, 1825–1833 (2010)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Parmar, R.K., Chopra, P., Paris, R.B.: On an extension of extended beta and hypergeometric functions. J. Class. Anal. 11, 91–106 (2017)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Rahman, G., Nisar, K.S., Tomovski, Z.: On a certain extension of the Riemann–Liouville fractional derivative operator. Commun. Korean Math. Soc. 34, 507–522 (2019)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Shadab, M., Khan, M.F., Luis, J., Lopez-Bonilla, J.: A new Riemann–Liouville type fractional derivative operator and its application in generating functions. Adv. Differ. Equ. 2018, 167 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Srivastava, H.M., Agarwal, P.: Certain fractional integral operators and the generalized incomplete hypergeometric functions. Appl. Appl. Math. 8(2), 333–345 (2013)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Wang, J., Zhou, Y., O’Regan, D.: A note on asymptotic behaviour of Mittag-Leffler. Integral Transforms Spec. Funct. 29, 81–94 (2017)

    MathSciNet  Article  Google Scholar 

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Correspondence to Meenakshi Singhal or Ekta Mittal.

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Singhal, M., Mittal, E. On a \(\psi \)-Generalized Fractional Derivative Operator of Riemann–Liouville with Some Applications. Int. J. Appl. Comput. Math 6, 143 (2020). https://doi.org/10.1007/s40819-020-00892-5

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Keywords

  • \(\psi \)-Gamma function
  • \(\psi \)-Beta function
  • \(\psi \)-Hypergeometric function
  • \(\psi \)-Generalized Riemann–Liouville fractional derivative operator
  • Mellin transform
  • Generating function

Mathematics Subject Classification

  • 26A33
  • 33B15
  • 33C05
  • 33C65