The Ramanujan Journal

, Volume 48, Issue 1, pp 21–32 | Cite as

Generalization of Taylor’s formula and differential transform method for composite fractional q-derivative

  • Lata Chanchlani
  • Subhash Alha
  • Jaya GuptaEmail author


In the present paper, we first establish a generalized q-Taylor’s formula involving composite fractional q-derivative. Next, we define the generalized q-differential transform and its inverse for composite fractional q-derivative and establish some basic properties for this transform. We also illustrate the effectiveness of these results by solving two fractional q-difference equations.


q-Taylor’s formula Composite fractional q-Derivative q-Differential transform method q-Difference equation 

Mathematics Subject Classification

26A33 33E12 34A12 35A22 33D90 



The support provided through UGC-Minor Research Project under XII plan grant of Maulana Azad National Urdu University, Hyderabad is gratefully acknowledged. The authors are grateful to Prof. Mridula Garg for inspiring discussions and helpful comments during the preparation of the paper. The authors are also thankful to the anonymous referee for the fruitful suggestions which led to the present form of paper.


  1. 1.
    Agarwal, R.P.: Certain fractional \(q\)-integrals and \(q\)-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Al-Salam, W.A.: Some fractional \(q\)-integrals and \(q\)-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Annaby, M.H., Mansour, Z.S.: \(q\)-Fractional Calculus and Equations. Springer, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Garg, M., Chanchlani, L., Alha, S.: On generalized \(q\)-differential transform. Aryabhatt J. Math. Inform. 5(2), 265–274 (2013)zbMATHGoogle Scholar
  5. 5.
    Hassan, H.A.: Generalized q-Taylor formula. Adv. Differ. Equ. 1, 1–2 (2016)MathSciNetGoogle Scholar
  6. 6.
    Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Jing, S.C., Fan, H.Y.: \(q\)-Taylor’s formula with its \(q\)-remainder. Commun. Theor. Phys. 23(1), 117–120 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Odibat, Z., Momani, S., Erturk, V.S.: Generalized differential transform method: application to differential equations of fractional order. Appl. Math. Comput. 197, 467–477 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Rajković, P.M., Marinković, S.D., Stanković, M.S.: On \(q\)-analogues of Caputo derivatives and Mittag-Leffler function. Fract. Calc. Appl. Anal. 10(4), 359–374 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rajković, P.M., Marinković, S.D., Stanković, M.S.: Fractional integrals and derivatives in \(q\)-calculus. Appl. Anal. Discret. Math. 1, 311–323 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zhou, J.K.: Differential Transformation and Its Applications for Electrical Circuits. Huazhong University Press, Wuhan (1986)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Science and HumanitiesGovt. R. C. Khaitan Polytechnic CollegeJaipurIndia
  2. 2.Department of MathematicsMaulana Azad National Urdu UniversityHyderabadIndia
  3. 3.Department of MathematicsJ. K. Lakshmipat UniversityJaipurIndia

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