Skip to main content
SpringerLink
Log in

q-Difference Recurrence Relations of Aleph Function with Generalization to nth Derivative

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Sharma and Jain studied the basic analogue of Meijer’s G-function by using the methods of q-calculus and constructed the q-difference operators and their lie algebra. In this article, we employ the same technique for basic anlogue of the Aleph function defined by Dutta et al. and thus on account of general nature of q-analogue of Aleph function,the q-difference recurrence relations of the Aleph function are obtained and an enumerous number of new and known results are special cases of our main results. Also the recurrence relation of I-function and Fox-H function are deduced as special cases by suitable substitutions. Furthermore the generalisation to nth partial derivatives and the required relations are also discussed which are quite general in nature and capable of producing a number of new and known results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

The data is provided on the request to the authors.

References

  1. Salam, A., Verma, W.A.: A Fractional Leibniz q-formula. Pac. J. Math. 60, 1–9 (1975)

    Article  MathSciNet  Google Scholar 

  2. Sdland, N., Banumann, B., Nonnenmadier, T.F.: Fractional Driftless Fokker -Planck equation with Power Law Diffusion Coefficients in. Springer, Berlin, pp. 513–525 (2001)

  3. Gangha, V.G., Mayr, E.W., Vorozhtsov, W.G.: Computer Algebra in Scientific Computing CASC Konstanz (2001)

  4. Saxena, R.K., Pogany, T.K.: On fractional integration formulae for aleph functions. Appl. Math. Comput. 218, 985–990 (2011)

    MathSciNet  Google Scholar 

  5. Ahmad, A., Jain, R., Jain, D.K.: \(q-\)analogue of Aleph function and its transformation fomulae with \(q-\)derivative. J. Stat. Appl. 3, 567–575 (2017)

    Article  Google Scholar 

  6. Kac, V.G., Cheung, P.: Quantum Calculus. Universitext Springer, New York (2002)

    Book  Google Scholar 

  7. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  8. Dutta, B.K., Arora, L.K.: On a basic anlogue of generalized H-function. Int. J. Math. Eng. sci. 1, 21–30 (2011)

    Google Scholar 

  9. Askey, R.: q-gamma and q-beta Functions. Appl. Anal. 8, 125–141 (1975)

    Article  MathSciNet  Google Scholar 

  10. Jain, D.K., Jain, R., Ahmad, F.: Some transformation formulae for basic analogue of I-function. Asian J. Math. Stat. 5(4), 158–162 (2011)

    Google Scholar 

  11. Sharma, R.J.: Lie Theoritic origin of basic analogue of Miejer’s G-function. J. Indian. S. 79, 173–183 (2012)

    Google Scholar 

  12. Agarwal, R.P.: Certain fractional \(q-\)integrals and \(q-\)derivatives. Proc. Edin. Math. 66, 365–370 (1969)

    Article  MathSciNet  Google Scholar 

  13. Saxena, R.K., Pogany, T.K.: Mathieu-type series for the aleph-function occuring in Fokker–Planck equation. Eur. J. Pure Appl. Math. 3, 980–988 (2010)

    MathSciNet  Google Scholar 

  14. Salam, A., Verma, W.A.: Some Fractional q-integrals and \(q-\)derivatives. Proc. Edinb. Math. 15, 135–140 (1966)

    Article  MathSciNet  Google Scholar 

  15. Kumar, D., Ayant, F., Sundland, N., Choi, J: Certain q- Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of Several Variable Aleph-function, Axioms (2023)

  16. Jackson, F.H.: On q-definite integrals. In Quart. J. Pure Appl. Math. 41, 193–203 (1910)

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Research Grant (No. JKST &IC/SRE/J/357-60) provided by JKSTIC, UT. of Jammu and Kashmir,India.

Funding

No funding was received for this work

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Islamic University of Science and Technology, Awantipora, Kashmir, India

    Mohammad Younus Bhat & Humaira Maqbool

  2. Mathematics and Computing Skills, Preparatory Studies Centre, University of Technology and Applied Sciences, Salalah, Sultanate of Oman

    Altaf A. Bhat

Authors
  1. Mohammad Younus Bhat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Humaira Maqbool
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Altaf A. Bhat
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors equally contributed towards this work.

Corresponding author

Correspondence to Mohammad Younus Bhat.

Ethics declarations

Conflict of interest

The authors have no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhat, M.Y., Maqbool, H. & Bhat, A.A. q-Difference Recurrence Relations of Aleph Function with Generalization to nth Derivative. Int. J. Appl. Comput. Math 10, 85 (2024). https://doi.org/10.1007/s40819-024-01724-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-024-01724-6

Keywords

Mathematics Subject Classification

Search

Navigation