1,532
Views
1
CrossRef citations to date
0
Altmetric
Applied & Interdisciplinary Mathematics

Generalized extended Mittag-Leffler function and its properties pertaining to integral transforms and fractional calculus

, & | (Reviewing editor:)
Article: 2220205 | Received 11 Nov 2022, Accepted 28 May 2023, Published online: 21 Jun 2023

References

  • Agarwal, R. P. (1953). A propos d’une Note M. Pierre Humbert. (French). Comptes rendus hebdomadaires des séances de l’Académie des sciences, 236, 2031–2032.
  • Agarwal, P., Mdallal, Q. A., Cho, Y. J., & Jain, S. (2018). Fractional differential equations for the generalized Mittag-Leffler function. Adv. Difference Equ.Advances in Difference Equations, 2018(1), 8. Paper No. 58. https://doi.org/10.1186/s13662-018-1500-7
  • Agarwal, P., Suthar, D. L., Jain, S., & Momani, S. (2022). The extended multi-iIndex Mittag-Leffler functions and their properties connected with fractional calculus and integral transforms. Thai Journal of Mathematics, 20(3), 1251–1266.
  • Attiya, A. A., Seoudy, T. M., & Albaid, A. (2023). Third-order differential subordination for meromorphic functions associated with generalized Mittag-Leffler function. Fractal fract.Fractal and Fractional, 7(2), 175. https://doi.org/10.3390/fractalfract7020175
  • Bayrak, M. A., & Demir, A. (2022). On the challenge of identifying space dependent coefficient in space-time fractional diffusion equations by fractional scaling transformations method. Turkish Journal of Science, 7(2), 132–145.
  • Choi, J., Kachhia, K. B., Prajapati, J. C., & Purohit, S. D. (2016). Some integral transforms involving extended generalized Gauss hypergeometric functions. Communication Korean Mathematical Society, 31(4), 779–790.
  • Dzherbashyan, M. M. (1966). Integral transforms and representations of functions in the complex domain. Izdat, Nauka.
  • Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. G. (1954). Higher transcendental functions (Vol. 2). Mc Graw-Hill.
  • Erdélyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. G. (1955). Higher transcendental functions (Vol. III). McGraw-Hill.
  • Gorenflo, R., Kilbas, A. A., & Rogosin, S. V. (1988). On the generalized Mittag-Leffler type function. Integral Transforms and Special Functions, 7(3–4), 215–224. https://doi.org/10.1080/10652469808819200
  • Gupta, R. K., Shaktawat, B. S., & Kumar, D. (2016). Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function. Journal of Rajasthan Academy Physical Science, 15(3), 117–126.
  • Humbert, P., & Agarwal, R. P. (1953). Sur la fonction de Mittag-LefferLeffler et quelques unes de ses generalizations. Bulletin des Sciences Mathematiques, 77(2), 180–185.
  • Jaimini, B. B., Sharma, M., Suthar, D. L., & Purohit, S. D., Hammouch, Z. (2021). On multi-index Mittag–Leffler function of several variables and fractional differential equations. Journal of Mathematics, 2021, 1–8. 2021, Art. ID 5458037. https://doi.org/10.1155/2021/5458037
  • Kilbas, A. A., & Saigo, M. (1995). Fractional integrals and derivatives of Mittag-Leffler type function (Russian). Doklady Akademii Nauk Belarusi, 39(4), 22–26.
  • Kilbas, A. A., & Saigo, M. (1996). On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations. Integral Transforms and Special Functions, 4(4), 355–370. https://doi.org/10.1080/10652469608819121
  • Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. In North-Holland mathematics studies (Vol. 204). Elsevier Science B.V. Amsterdam.
  • Kumar, D., Ayant, F. Y., Nirwan, P., & Suthar, D. L., Srivastava, H. M. (2022). Boros integral involving the generalized multi-index Mittag-Leffler function and incomplete I-functions. Research in Mathematics, 9(1), 7. Paper No. 2086761. https://doi.org/10.1080/27684830.2022.2086761
  • Kumar, D., & Purohit, S. D. (2014). Fractional differintegral operators of the generalized Mittag-Leffler type function. Malaya J. Mat.Malaya Journal of Matematik, 2(4), 419–425. https://doi.org/10.26637/mjm204/008
  • Kumar, D., Ram, J., & Choi, J. (2022). Dirichlet averages of generalized Mittag-Leffler type function. Fractal and Frac.Fractal and Fractional, 6(6), 297, 112297, 112297, 1–12. https://doi.org/10.3390/fractalfract6060297
  • Mathai, A. M., Saxena, R. K.,& Haubold, H. J. (2010). The H-function Theory and Application. Springer. https://doi.org/10.1007/978-1-4419-0916-9
  • Mishra, V. N., Suthar, D. L., & Purohit, S. D., Srivastava, H. M. (2017). Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function. Cogent Mathematics, 4(1), 11. Art. ID 1320830. https://doi.org/10.1080/23311835.2017.1320830
  • Mittag Leffler, G. M. (1903). Sur la nouvelle fonction Eα(x). Comptes rendus hebdomadaires des séances de l’Académie des sciences, 137, 554–558.
  • Nisar, K. S., Suthar, D. L., Agarwal, R., & Purohit, S. D. (2020). Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag-Leffler function. Adv. Difference Equ.Advances in Difference Equations, 2020(1), 14. Paper No. 148. https://doi.org/10.1186/s13662-020-02610-3
  • Özarslan, M. A., & Yilmaz, B. (2014). The extended Mittag-Leffler function and its properties. Journal of Inequalities and Applications, 2014(1), 10. 2014. https://doi.org/10.1186/1029-242X-2014-85
  • Parmar, R. K. (2015). A class of extended Mittag–Leffler functions and their properties related to integral transforms and fractional calculus. Mathematics, 3(4), 1069–1082. https://doi.org/10.3390/math3041069
  • Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Mathematical Journal, 19, 7–15.
  • Purohit, S. D., Kalla, S. L., & Suthar, D. L. (2011). Fractional integral operators and the multiindex Mittag-Leffler functions. Science of Serial A Mathematics Science (NS), 21, 87–96.
  • Salim, T. O. (2009). Some properties relating to generalized Mittag-Leffler function. Advances in ApplliedApplied Mathematical Analysis, 4(1), 21–30.
  • Sneddon, I. N. (1979). The use of integral transform. Tata Mc Graw Hill.
  • Srivastava, H. M., Parmar, R. K., & Chopra, P. (2012). A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions. Axioms, 1(3), 238–258. https://doi.org/10.3390/axioms1030238
  • Suthar, D. L., & Amsalu, H. (2017). Generalized fractional integral operators involving Mittag-Leffler function. Applications and Applied Mathematics: An International Journal, 12(2), 1002–1016. https://doi.org/10.1155/2018/7034124
  • Suthar, D. L., Amsalu, H., & Godifey, K. (2019). Certain integrals involving multivariate Mittag-Leffler function. J. Inequal. Appl.Journal of Inequalities and Applications, 2019(1), 16. Paper No.r̥ 208. https://doi.org/10.1186/s13660-019-2162-z
  • Suthar, D. L., Andualem, M., & Debalkie, B. (2019). A study on generalized multivariable Mittag-Leffler function via generalized fractional calculus operators. J. Math.Journal of Mathematics, 2019, 1–7. 2019, Art. ID 9864737. https://doi.org/10.1155/2019/9864737
  • Suthar, D. L., & Purohit, S. D. (2014). Unified fractional integral formulae for the generalized Mittag-Leffler functions. Journal of Science Arts, 27(2), 117–124.
  • Suthar, D. L., Shimelis, B., Abeye, N., & Amsalu, H. (2020). New composition formulae for the generalized fractional calculus operators with the extended Mittag-Leffler function. Mathematics in Engineering, Science and Aerospace, 11(2), 309–321.
  • Whittaker, E. T., & Watson, G. N. (1962). A course of modern analysis. Cambridge University Press.
  • Wiman, A. (1905). Über den Fundamental Satz in der Theorie der Functionen Eα(x). Acta Mathematica, 29, 191–201.
  • Zhang, T., & Li, Y. (2022). S-asymptotically periodic fractional functional differential equations with off-diagonal matrix Mittag-Leffler function kernels. Math. Comput. SimulationMathematics and Computers in Simulation, 193, 331–347. https://doi.org/10.1016/j.matcom.2021.10.006