Skip to main content
SpringerLink
Log in

A Class of the Extended Multi-index Bessel-Maitland Functions and It’s Properties

  • Chapter
  • First Online:
Mathematical and Computational Intelligence to Socio-scientific Analytics and Applications

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 518))

  • 148 Accesses

Abstract

Here, we present the extended multi-index Bessel-Maitland function (EMBMF) defined using the extended beta function and investigate its several properties including, integral representation, derivatives, beta transform and Mellin transform. The relationships of the said function (EMBMF) with the Laguerre polynomial and Whittaker function, respectively, are also elaborated. Further, image formulas for the extended multi-index Bessel-Maitland function are also investigated using fractional calculus operators. All the derived results generalize plentiful well-known results and can be used to derive a number of pertinent results in the theory of special functions.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Marichev OI (1983) Handbook of integral transform and Higher transcendental functions. Ellis Harwood Chichester, New York

    MATH  Google Scholar 

  2. Watson GN (1965) A treatise on the theory of Bessel functions. Cambridge Mathematical Library edition Cambridge University Press

    Google Scholar 

  3. Pathak RS (1966) Certain convergence theorems and asymptotic properties of a generalization of Lommel and Maitland transform. Proc Nat Acad Sci India Sect A 36:81–86

    MathSciNet  MATH  Google Scholar 

  4. Choi J, Agarwal P (2016) A note on fractional integral operator associated with multi-index Mittag-Leffler. Filomat 30(70):1931–1939

    Article  MathSciNet  MATH  Google Scholar 

  5. Agarwal R, Jain S, Agarwal RP, Baleanu D (2018) A remark on the fractional integral operators and the image formulas of generalized Lommel-Wright function. Front Phys 6:79. https://doi.org/10.3389/fphy.2018.00079

    Article  Google Scholar 

  6. Nisar KS, Purohit SD, Suthar DL, Singh J (2019) Fractional order integration and certain integrals of generalized multiindex Bessel function. In: Singh J., Kumar D., Dutta H., Baleanu D. Purohit S. (eds) Mathematical Modelling, Applied Analysis and Computation. ICMMAAC 2018. Springer Proceedings in Mathematics & Statistics 272 Springer, Singapore

    Google Scholar 

  7. Purohit SD, Kalla SL, Suthar DL (2011) Fractional integral operators and the multi-index Mittag-Leffler functions. Sci Ser A Math Sci (NS) 21:87–96

    MATH  Google Scholar 

  8. Suthar DL, Amsalu H (2017) Certain integrals associated with the generalized Bessel-Maitland function. Appl Appl Math 12(2):1002–1016

    MathSciNet  MATH  Google Scholar 

  9. Suthar DL, Khan AM, Alaria A, Purohit SD, Singh J (2020) Extended Bessel-Maitland function and its properties pertaining to integral transforms and fractional calculus. AIMS Mathematics 5(2):1400–1410

    Article  MathSciNet  MATH  Google Scholar 

  10. Suthar DL, Purohit SD, Parmar RK (2017) Generalized fractional calculus of the multiindex Bessel function. Math Nat Sci 1:26–32

    Article  Google Scholar 

  11. Suthar DL, Tsagye T (2017) Riemann-Liouville fractional integrals and differential formula involving multiindex Bessel-function. Math Sci Lett 6(3):1–5

    Article  Google Scholar 

  12. Suthar DL, Habenom H, Tadesse H (2018) Generalized fractional calculus formulas for a product of Mittag-Leffler function and multivariable polynomials. Int J Appl Comput Math 4(1):1–12

    Article  MathSciNet  MATH  Google Scholar 

  13. Tilahun K, Tadessee H, Suthar DL (2020) The Extended Bessel-Maitland Function and Integral Operators Associated with Fractional Calculus. Journal of Mathematics 7582063:1–8

    Article  MathSciNet  MATH  Google Scholar 

  14. Albayrak, D, Dernek, A, Dernek, N, Uçar, F (2021) New integral transform with generalized Bessel-Maitland function kernel and its applications. Math Meth Appl Sci 44: 1394-1408. https://doi.org/10.1002/mma.6837

  15. Prabhakar TR (1971) A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J 19:7–15

    MathSciNet  MATH  Google Scholar 

  16. Shukla AK, Prajapati JC (2007) On a generalized Mittag-Leffler function and its properties. J Math Anal Appl 336:797–811

    Article  MathSciNet  MATH  Google Scholar 

  17. Agarwal P, Jain S, Agarwal S, Nagpal M (2014) On a new class of integrals involving Bessel functions of the first kind. Commun Numer Anal Art ID 00216:1–7

    Google Scholar 

  18. Choi J, Agarwal P, Mathur S, Purohit SD (2014) Certain new integral formulas involving the generalized Bessel functions. Bull Korean Math Soc 51(4):995–1003

    Article  MathSciNet  MATH  Google Scholar 

  19. Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (1955) Higher transcendental functions, vol 3. Mc Graw-Hill, New York

    MATH  Google Scholar 

  20. Olver FWL, Lozier DW, Boisvert RF, Clark CW (2010) NIST handbook of mathematical functions. Cambridge University Press

    Google Scholar 

  21. Saxena RK, Nishimoto K (2010) N-fractional calculus of generalized Mittag-Leffler functions. J Fract Calc 37:43–52

    MathSciNet  MATH  Google Scholar 

  22. Parmar RK (2015) A class of extended Mittag-Leffler functions and their properties related to integral transforms and fractional calculus. Mathematics 3:1069–1082

    Article  MathSciNet  MATH  Google Scholar 

  23. Wiman A (1905) Über der Fundamentalsatz in der Theorie der Funktionen \(E_{\alpha }(x)\). Acta Math 29:191–201

    Google Scholar 

  24. Mittag-Leffler GM (1903) Sur la nouvelle fonction \(E_{\alpha }(x)\). CR Acad Sci 137:554–558

    Google Scholar 

  25. Sneddon IN (1979) The use of integral transform. Tata McGraw Hill, New Delhi

    Google Scholar 

  26. Srivastava HM, Karlson PW (1985) Multiple Gaussian hypergeometric series. Halsted Press, New York

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Özarslan MA (2012) Some remarks on extended hypergeometric, extended confluent hypergeometric and extended Appell’s functions. J Comput Anal Appl 14(6):1148–1153

    MathSciNet  MATH  Google Scholar 

  29. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North-Holland mathematical studies, Elsevier, London

    MATH  Google Scholar 

  30. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Switzerland

    MATH  Google Scholar 

  31. Hilfer REd (2000) Applications of fractional calculus in physics. World Scientific Publishing Company, Singapore

    Book  MATH  Google Scholar 

  32. Srivastava HM, Tomovski Z (2009) Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernel. Appl Math Comput 211(1):198–210

    MathSciNet  MATH  Google Scholar 

  33. Mittal E, Pandey RM, Joshi S (2016) On extension of Mittag-Leffler function. Appl Appl Math 11(1):307–316

    MathSciNet  MATH  Google Scholar 

  34. Özarslan MA, Yilmaz B (2014) The extended Mittag-Leffler function and its properties. J Inequal Appl. https://doi.org/10.1186/1029-242X-2014-85

    Article  MathSciNet  MATH  Google Scholar 

  35. Saxena RK, Ram J, Suthar DL (2009) Generalized fractional calculus of the generalized Mittag-Leffler functions. J Indian Acad Math 31(1):165–172

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Wollo University, P. O. Box: 1145, Dessie, Dessie, Ethiopia

    D. L. Suthar

  2. Department of Physics, Wollo University, P. O. Box: 1145, Dessie, Dessie, Ethiopia

    Mangesha Ayene

  3. Sur University College, P. O. Box: 440, Sur, P. C. 411, Sultanate of Oman

    V. K. Vyas & Ali A. Al-Jarrah

Authors
  1. D. L. Suthar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mangesha Ayene
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. K. Vyas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ali A. Al-Jarrah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. L. Suthar .

Editor information

Editors and Affiliations

  1. Department of Mathematics, MNNIT Allahabad, FATER Academy of India, Prayagraj, India

    Pankaj Srivastava

  2. School of Mathematics, Madurai Kamaraj University, Madurai, India

    M. Lellis Thivagar

  3. Department of Mathematics and Computer Science, University of Oradea, Oradea, Romania

    Georgia Irina Oros

  4. Rattanakosin International College of Creative Entrepreneurship, Rajamangala University of Technology, Nakhon Pathom, Thailand

    Chai Ching Tan

Ethics declarations

The authors declare that there is no conflict of interests regarding the publication of this paper.

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Suthar, D.L., Ayene, M., Vyas, V.K., Al-Jarrah, A.A. (2022). A Class of the Extended Multi-index Bessel-Maitland Functions and It’s Properties. In: Srivastava, P., Thivagar, M.L., Oros, G.I., Tan, C.C. (eds) Mathematical and Computational Intelligence to Socio-scientific Analytics and Applications. Lecture Notes in Networks and Systems, vol 518. Springer, Singapore. https://doi.org/10.1007/978-981-19-5181-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation