Analytical Insights into the Modified Fractional Bell Polynomial with Mittag-Leffler Parameter

Bhaktaraj Thiyam

Department of Mathematics, Manipur International University, Imphal, India

Md. Indraman Khan

Department of Mathematics, PETTIGREW College, Ukhrul, Manipur University, Manipur


Vol: 14, Issue: 2, 2024

Receiving Date: 2024-02-29 Acceptance Date:


Publication Date:


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In this paper, we will define the Modified Fractional Bell Polynomial by incorporating the Mittag Leffler function of one parameter. The Existence and convergence of the Modified Fractional Bell Polynomial will be established by extending the classical results of Bell Polynomial and Mittag Leffler function of one parameter in the fractional calculus. Additionally, we explore the inverse of the Modified Fractional Bell Polynomial, providing a step-by-step proof of its existence. This result enhances the applicability of the polynomial by allowing a unique mapping from each output to a set of input values. The introduction of the Modified Fractional Bell Polynomial, with its well-established properties, opens avenues for further research and applications in diverse mathematical contexts. The generality of the polynomial makes it a powerful tool for modeling complex phenomena.

Keywords: Fractional Bell Polynomials; Mittag-Leffler Function; Generalization; Modified Fractional Bell Polynomial; Existence; Convergence; Inverse Function; Mathematical Analysis; Special Functions


  1. Nishimoto, K., An essence of Nishimoto’s Fractional Calculus, Descartes Press Co.1991.
  2. B. Ross, A brief history and exposition of the fundamental theory of fractional calculus, In: Fractional Calculus and Its Applications. Ed. by B. Ross., Lecture Notes in Mathematics. Springer Berlin Heidelberg, 457(1975), 1-36.
  3. B. Ross. Fractional calculus. In: Mathematics Magazine 50(3) (1977), 115-122.
  4. B. Ross, The development of fractional calculus 1695-1900. In: Historia Mathematica, 4(1) (1977), 75-89.
  5. P.L. Butzer and U. Westphal. An introduction to fractional calculus. In: Applications of Fractional Calculus in Physics. Ed. by R. Hilfer. World Scientific, Chap. 3(2000).
  6. K. Oldham and J. Spanier.The Fractional Calculus. Theory and Applications of Differentiation and Integration toArbitrary Order. Academic Press, INC, San Diego Ca, 1974.
  7. K. S. Miller and B. Ross.An Introduction to the Fractional Calculus and Fractional Differential Equations. JohnWiley & Sons, Inc., New York, NY, 1993.
  8. S. Samko, A. Kilbas, and O. Marichev.Fractional Integrals and Derivatives: Theory and Applications. Gordon andBreach, London, 1993.
  9. I. Podlubny.Fractional Differential Equations. Academic Press, INC, San Diego Ca, 1999.
  10. K. Diethlem. The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer-Verlag, Berlin, Heidelberg, 2010.
  11. D. Baleanu, K. Diethlem, E. Scalas, and J.J. Trujillo. Fractional Calculus. Models and Numerical Methods. World Scientific, Singapore, 2012.
  12. A. A. Kilbas, M. Srivastava H, and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier Science, 2006.
  13. F. Mainardi. Fractional Calculus and Waves in Linear Viscoelasticy. Imperial Collage Press, London, 2010.
  14. Y. Povstenko. Fractional Thermoelasticy. Springer International Publishing, Cham, Heidelberg, New York, Dodrecht, London, 2015.
  15. Y. Povstenko. Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkhuser, Springer, Cham, Heidelberg, New York, Dodrecht, London, 2015.
  16. Riordan, J. (1968). Combinatorial Identities. Wiley. Chapter 5: ”Bell Numbers and Their Exponential Generating Functions.”
  17. Bell, E. T. (1934). ”Exponential Polynomials.” Annals of Mathematics, 35(2), 258-277.
  18. A.M. Mathai, R.K. Saxena The -function with Applications in Statistics and Other Disciplines Wiley East. Ltd., New Delhi (1978)
  19. H.M. Srivastava, B.R.K. Kashyap Special Functions in Queuing Theory and Related Stochastic Processes Acad. Press, New York (1981)
  20. H.M. Srivastava, K.C. Gupta, S.P. Goyal The -Functions of One and Two Variables with Applications South Asian Publs., New Delhi (1982)
  21. A. Prudnikov, Yu. Brychkov, O. Marichev Integrals and Series, Some More Special Functions, vol. 3, Gordon & Breach, New York (1992)
  22. V. Kiryakova Generalized Fractional Calculus and Applications Longman & J. Wiley, Harlow, New York (1994)
  23. A.M. Mathai, H.J. Haubold Special Functions for Applied Scientists Springer (2008)
  24. Richard Askey. The q -gamma and q -beta functions. Applicable Anal., 8(2):125-141, 1978/79.
  25. Richard Askey. Ramanujan’s extensions of the gamma and beta functions. Amer. Math. Monthly, 87(5):346-359, 1980.
  26. M. Aslam Chaudhry and S. M. Zubair. Generalized incomplete gamma functions with applications. J. Comput. Appl. Math., 55(1):99-124, 1994.
  27. M. Aslam Chaudhry and S. M. Zubair. On the decomposition of generalized incomplete gamma functions with applications of Fourier transforms. J. Comput. Appl. Math., 59(3):253-284, 1995.
  28. Allen R. Miller. Reductions of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals. Rocky Mountain J. Math., 30(2):703-714, 2000.
  29. Larry C. Andrews. Special functions for engineers and applied mathemati- cians. Macmillan Co., New York, 1985.
  30. M. Aslam Chaudhry, Asghar Qadir, M. Raque, and S. M. Zubair. Extension of Euler’s beta function. J. Comput. Appl. Math., 78(1):19-32, 1997.
  31. Agarwal R.P. ,A propos d’une note M. Pierre Humbert,C.R.Acad.Sci. Paris 236 (1953),pp.2031-2032.
  32. Chouhan Amit and Satishsaraswat, Some Rmearks on Generalized Mittag-Leffler Function and Fractional operators ,IJMMAC Vol2, No.2, pp. 131-139.
  33. GM.Mittag - Leffler , Sur la nouvelle function of Eα(x) ,C.R.Acad. Sci. Paris 137 (1903), pp.554-558.
  34. Humbert P. and Agarwal R.P., Sur la function de Mittag - Leffler et quelquesunes deses generalizations, Bull. Sci.Math. (2)77(1953), pp.180-186.
  35. Mohammed Mazhar-ul- Haque and Holambe T.L. , A Q function in fractional calculus, Journal of Basic and Applied Research International, International knowledge press, vol. 6, issue 4 (2015),pp.248-252.
  36. Shukla A.K. and Prajapati J.C., On a generalization of Mittag - Leffler function and its properties, J.Math.Anal.Appl.336(2007),pp.79-81.
  37. Salim T.O. and Faraj O., A generalization of Mittag-Leffler function and Integral operator associated with the Fractional calculus, Journal of Fractional Calculus and Applications,3(5) (2012),pp.1-13.
  38. Wiman A., Uber de fundamental satz in der theorie der funktionen, Acta Math. 29(1905),pp.191-201.
  39. T. R. Prabhakar, A singular integral equation with a generalized Mittal-Leffler function in the Kernel, Yokohama Math.J.,19(1971), 7-15.
  40. P. Delerue Sur le calcul symbolique variables et fonctions hyperbesseliennes (II) Annales Soc. Sci. Bruxelles, 1, 3 (1953), pp. 229-274
  41. Herbert S. Wilf, “Generating functionology,” Academic Press, 1994.
  42. Ronald L. Graham, Donald E. Knuth, Oren Patashnik, “Concrete Mathematics,” AddisonWesley, 1994.
  43. Ian P. Goulden, David M. Jackson, “Combinatorial Enumeration,” John Wiley and Sons, 1983.
  44. J. H. van Lint, R. M. Wilson, “A Course in Combinatorics,” Cambridge University Press, 2001.

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