On a Generalized Mittag-Leffler Function and Fractional Integrals

Year 2024, , 12 - 25, 31.03.2024

Virendra Kumar

https://doi.org/10.33401/fujma.1378534

Abstract

The object of this paper is to study a generalized Mittag-Leffler function and a modified general class of functions which is reducible to several special functions. convergent conditions of these functions are discussed. Some results pertaining to the generalized Mittag-Leffler function and generating relations involving these functions are derived. Further, fractional integrals involving these functions are achieved. Some illustrative exclusive cases of the results are presented.

Keywords

Mittag-Leffler function, Riemann-Liouville fractional integrals, special functions

References

• [1] E.W. Barnes, The asymptotic expansion of integral functions defined by Taylor’s series, Philos. Trans. Roy. Soc. London Ser. A Math. Phys. Sci., 206 (1906), 249-297.
• [2] E.M. Wright, The asymptotic expansion of integral functions defined by Taylor series, I. Philos. Trans. Roy. Soc. London Ser. A Math. Phys. Sci., 238 (1940), 423-451.
• [3] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
• [4] H.M. Srivastava, An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions, J. Adv. Engrg. comput., 5(3) (2021), 135-166. $\href{http://dx.doi.org/10.55579/jaec.202153.340}{[\mbox{CrossRef}]}$
• [5] V. Kumar, On the generalized Hurwitz-Lerch zeta function and generalized Lambert transform, J. Classical Anal., 17 (1) (2021), 55–67. $\href{http://dx.doi.org/10.7153/jca-2021-17-05}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85132267417&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+generalized+Hurwitz-Lerch+zeta+function+and+generalized+Lambert+transform%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=0}{[\mbox{Scopus}]} %\href{}{[\mbox{Web of Science}]}$
• [6] G.M. Mittag-Leffler, Sur la nouvelle function Ea (x), C. R. Acad. Sci. Paris, 137 (1903), 554-558.
• [7] A. Wiman, Uber den Fundamentalsatz in der Teorie der Funktionen Ea (x), Acta. Math., 29 (1905), 191-201.
• [8] P. Humbert and R.P. Agarwal, la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bull. Sci. Math., 77(2) (1953), 180-185.
• [9] N. Khan, M. Aman and T. Usman, Extended Beta, hypergeometric and confluent hypergeometric functions via multi-index Mittag-Leffler function, Proc. Jang. Math. Soc., 25(1) (2022), 43-58. $\href{http://dx.doi.org/10.17777/pjms2022.25.1.43}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85130358565&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Extended+Beta%2C+hypergeometric+and+confluent+hypergeometric+functions+via+multi-index+Mittag-Leffler+function%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=0}{[\mbox{Scopus}]}$
• [10] M. Kamarujjama, N.U. Khan and O. Khan, Extended type k-Mittag-Leffler function and its applications, Int. J. Appl. Comput. Math., 5(3) (2019), Article No. 72. $\href{https://doi.org/10.1007/s40819-019-0656-5}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85070418196&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Extended+type+k-Mittag-Leffler+function+and+its+applications%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=0}{[\mbox{Scopus}]} %\href{}{[\mbox{Web of Science}]}$
• [11] M.A. Khan and S. Ahmed, On some properties of the generalized Mittag-Leffler function, Springer Plus, 2 (2013), Article No. 337. $\href{https://doi.org/10.1186/2193-1801-2-337}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84881253775&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+some+properties+of+the+generalized+Mittag-Leffler+function%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000209465000135}{[\mbox{Web of Science}]}$
• [12] M.A. Khan, S. Ahmed, On some properties of fractional calculus operators associated with generalized Mittag-Leffler function, Thai J. Math., 3 (2013), 645-654. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84885450498&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+some+properties+of+fractional+calculus+operators+associated+with++generalized+Mittag-Leffler+function%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000416702200011}{[\mbox{Web of Science}]}$
• [13] N.U. Khan and M.I. Khan, Results concerning the analysis of generalized Mittag-Leffler function associated with Euler type integrals, Fasciculi Mathematici, 66 (2023), 49-59. $\href{http://dx.doi.org/10.21008/j.0044-4413.2023.0004}{[\mbox{CrossRef}]}$
• [14] N. Khan, M.I. Khan, T. Usman, K. Nonlaopon and S. Al-Omari Unified integrals of generalized Mittag-Leffler functions and their graphical numerical investigation, Symmetry, 14 (5) (2022), 869. $\href{https://doi.org/10.3390/sym14050869}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85129752996&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Unified+integrals+of+++generalized+Mittag-Leffler+functions+and+their+graphical+numerical+investigation%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000801507300001}{[\mbox{Web of Science}]} $
• [15] A.K. Shukla and J.C. Prajapati, On a generalized Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 97-112. $\href{https://doi.org/10.1016/j.jmaa.2007.03.018}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-34548078160&origin=resultslist&sort=plf-f&src=s&sid=81fa61a1532600f63071f79cd12df452&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+generalization+of+Mittag-Leffler+function+and+its+properties%22%29&sl=67&sessionSearchId=81fa61a1532600f63071f79cd12df452&relpos=2}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000249744000006}{[\mbox{Web of Science}]}$
• [16] T.O. Salim, Some properties relating to the generalized Mittag-Leffler function, Adv. Appl. Math. Anal., 4 (2009), 21-30.
• [17] H.M. Srivastava, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations, J. Nonlinear Convex Anal., 22(8) (2021), 1501-1520.
• [18] S.P. Goyal and R.K. Laddha, On the generalized Riemann zeta functions and the generalized Lambert transform, Ganita Sandesh, 11 (2) (1997), 99-108.
• [19] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, vol. I, McGraw-Hill Book Company, New York, 1953.
• [20] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of Integral Transforms, vol. II, McGraw-Hill Book Company, New York, 1954.
• [21] E.M. Wright, The asymptotic expansion of the generalized Bessel function, Proc. London Math. Soc., 38 (2) (1935), 257-270.
• [22] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, vol. II, McGraw-Hill Book Company, New York, 1953.