Relations Between the Fractional Operators in q-Calculus

  • Sergei Silvestrov
  • Predrag M. RajkovićEmail author
  • Sladjana D. Marinković
  • Miomir S. Stanković
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)


In this survey paper, we will consider the fractional operators in q-calculus. Starting from the fractional versions of q-Pochhammer symbol, we generalize the notions of the fractional q-integral and q-derivative by introducing variable lower bound of integration. We discuss their properties, describe relations which connect them, and illustrate notions and results with examples and counterexamples.


Fractional q-integral Fractional q-derivative 

MSC 2010 Classification

05A30 26A33 33D05 



This research was financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia. The second author is grateful to Mathematics and Applied Mathematics Research Environment at the School of Education, Culture and Communication at Mälardalen University for cordial hospitality and cooperation during his research stay.


  1. 1.
    Agarwal, R.P.: Certain fractional \(q\)-integrals and \(q\)-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Al-Salam, W.A.: Some fractional \(q\)-integrals and \(q\)-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1966)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Al-Salam, W.A.: \(q\)-Analogues of Cauchy’s formulas. Proc. Am. Math. Soc. 17(3), 616–621 (1966)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Al-Salam, W.A., Verma, A.: A fractional Leibniz \(q\)-formula. Pac. J. Math. 60(2), 1–9 (1975)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bangerezako, G.: Variational calculus on \(q\)-nonuniform lattices. J. Math. Anal. Appl. 306(1), 161–179 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gasper, G., Rahman, M.: Basic hypergeometric series. In: Encyclopedia of Mathematics and its Applications, 2nd edn., vol. 96. Cambridge University Press, Cambridge (2004)Google Scholar
  7. 7.
    Gauchman, H.: Integral inequalities in \(q\)-calculus. Comput. Math. Appl. 47, 281–300 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hahn, W.: Lineare Geometrische Differenzengleichungen, vol. 169. Berichte der Mathematisch-Statistischen Section im Forschungszentrum Graz (1981)Google Scholar
  9. 9.
    Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)CrossRefGoogle Scholar
  10. 10.
    Kiryakova, V.: Generalized Fractional Calculus and Applications (Pitman Research Notes in Mathematics. Series 301). Longman, Harlow & J. Wiley Ltd., New York (1999)Google Scholar
  11. 11.
    Koepf, W., Rajković, P.M., Marinković, S.D.: On a connection between formulas about \(q\)-gamma functions. J. Nonlinear Math. Phys. 23(3), 343–350 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley (1993)Google Scholar
  13. 13.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, Inc. (1974)Google Scholar
  14. 14.
    Rajković, P.M., Stanković, M.S., Marinković, S.D.: Mean value theorems in \(q\)-calculus. Matematički vesnik 54, 171–178 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Rajković, P.M., Marinković, S.D., Stanković, M.S.: Fractional integrals and derivatives in \(q\)-calculus. Appl. Anal. Discret. Math. 1, 311–323 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Stanković, M.S., Rajković, P.M., Marinković, S.D.: Inequalities which includes \(q\)-integrals. Bulletin (Acad. Serbe Sci. Arts. Cl. Sci. Math. Natur. Sci. Math.) 31, 137–146 (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Sergei Silvestrov
    • 1
  • Predrag M. Rajković
    • 2
    Email author
  • Sladjana D. Marinković
    • 3
  • Miomir S. Stanković
    • 4
  1. 1.Division of Applied Mathematics, School of Education, Culture and CommunicationMälardalen UniversityVästeråsSweden
  2. 2.Faculty of Mechanical Engineering, Department of MathematicsUniversity of NišNišSerbia
  3. 3.Faculty of Electronic Engineering, Department of MathematicsUniversity of NišNišSerbia
  4. 4.Faculty of Occupation Safety, Department of MathematicsUniversity of NišNišSerbia

Personalised recommendations