Geodesic
Bicomplex Mittag-Leffler function and associated properties
Agarwal, R.1 ; Sharma, U. P.2 ; Agarwal, R. P. 3
1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India
2 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, , India
3 Department of Mathematics, Texas A\(\&\)M University, Kingsville 700 University Blvd., Kingsville, USA
Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 1, p. 48-60.

Voir la notice de l'article provenant de la source International Scientific Research Publications

With the increasing importance of the Mittag-Leffler function in physical applications, these days many researchers are studying various generalizations and extensions of the Mittag-Leffler function. In this paper, efforts are made to define the bicomplex extension of the Mittag-Leffler function, and also its analyticity and region of convergence are discussed. Various properties of the bicomplex Mittag-Leffler function including integral representation, recurrence relations, duplication formula, and differential relations are established.
DOI : 10.22436/jnsa.015.01.04
Classification : 30G35, 33E12
Keywords: Bicomplex numbers, exponential function, Gamma function, Mittag-Leffler function

Agarwal, R. 1 ; Sharma, U. P. 2 ; Agarwal, R. P.  3

1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India
2 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, , India
3 Department of Mathematics, Texas A\(\&\)M University, Kingsville 700 University Blvd., Kingsville, USA 
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Agarwal, R.; Sharma, U. P.; Agarwal, R. P. . Bicomplex Mittag-Leffler function and associated properties. Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 1, p. 48-60. doi : 10.22436/jnsa.015.01.04. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.015.01.04/

[1] Agarwal, P.; Chand, M.; Baleanu, D.; O’Regan, D.; Jain, S. On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function, Adv. Difference Equ., Volume 2018 (2018), pp. 1-13 | DOI

[2] Agarwal, R.; Goswami, M. P.; Agarwal, R. P. Mellin transform in bicomplex space and its application, Stud. Univ. Babes¸-Bolyai Math., Volume 62 (2017), pp. 217-232 | DOI

[3] Agarwal, R.; Goswami, M. P.; Agarwal, R. P. Sumudu Transform In Bicomplex Space And Its Application, Ann. Appl. Math., Volume 33 (2017), pp. 239-253

[4] Agarwal, R.; Goswami, M. P.; Agarwal, R. P.; Venkataratnam, K. K.; Baleanu, D. Solution of Maxwell’s Wave Equations in Bicomplex Space, Romanian J. Phys., Volume 62 (2017), pp. 1-11

[5] Agarwal, P.; Nieto, J. J. Some fractional integral formulas for the Mittag-Leffer type function with four parameters, Open Math., Volume 13 (2015), pp. 537-546 | DOI

[6] Agarwal, P.; Rogosin, S. V.; Trujillo, J. J. Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions, Proc. Indian Acad. Sci. Math. Sci., Volume 125 (2015), pp. 291-306 | DOI

[7] Andrić, M.; Farid, G.; Pečarić, J. A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., Volume 21 (2018), pp. 1377-1395 | DOI

[8] Arshad, M.; Choi, J.; Mubeen, S.; Nisar, K. S.; Rahman, G. A new extension of the Mittag-Leffler function, Commun. Korean Math. Soc., Volume 33 (2018), pp. 549-560 | DOI

[9] Cockle, J. On a new Imaginary in Algebra, London Edinburgh Dublin Philos. Mag. J. Sci., Volume 34 (1849), pp. 37-47 | DOI

[10] Cockle, J.; Davis, T. S. On certain Functions resembling Quaternions, and on a new Imaginary in Algebra, London, Edinburgh Dublin Philos. Mag. J. Sci., Volume 33 (1848), pp. 435-439 | DOI

[11] Charak, K. S.; Rochon, D. On Factorization of Bicomplex Meromorphic Functions, In: Hypercomplex Analysis, Birkhauser Verlag, Basel, Volume 2009 (2009), pp. 55-68

[12] Charak, K. S.; Rochon, D.; Sharma, N. Normal Families of Bicomplex Holomorphic Functions, Fractals, Volume 17 (2009), pp. 257-268 | DOI

[13] Charak, K. S.; Rochon, D.; Sharma, N. Normal Families of Bicomplex Meromorphic Functions, Ann. Polon. Math., Volume 103 (2012), pp. 303-317 | DOI

[14] Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G. Higher transcendental functions, Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955

[15] Garraa, R.; Garrappa, R. The Prabhakar or three parameter Mittag-Leffler function: theory and application, Commun. Nonlinear Sci. Numer. Simul., Volume 56 (2018), pp. 314-329 | DOI

[16] Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S. V. Mittag-Leffler functions, Related Topics and Applications, Springer, Heidelberg, 2014 | DOI

[17] Goyal, R. Bicomplex polygamma function, Tokyo J. Math., Volume 30 (2007), pp. 523-530 | DOI

[18] Goyal, S. P.; Goyal, R. On bicomplex Hurwitz zeta function, South East Asian J. Math. Math. Sci., Volume 4 (2006), pp. 59-66

[19] Goyal, S. P.; Mathur, T.; Goyal, R. Bicomplex Gamma And Beta Function, J. Rajasthan Acad. Phys. Sci., Volume 5 (2006), pp. 131-142

[20] Haubold, H. J.; Mathai, A. M.; Saxena, R. K. Mittag-Leffler functions and their applications, J. Appl. Math., Volume 2011 (2011), pp. 1-51 | DOI

[21] Hucks, J. Hyperbolic complex structures in physics, J. Math. Phys., Volume 34 (1993), pp. 5986-6008 | DOI

[22] Irmak, H.; Agarwal, P. Some comprehensive inequalities consisting of Mittag-Leffler type functions in the complex plane, Math. Model. Nat. Phenom., Volume 12 (2017), pp. 65-71 | DOI

[23] Jain, S.; Agarwal, P.; Kilicman, A. Pathway fractional integral operator associated with 3m-parametric Mittag-Leffler functions, Int. J. Appl. Comput. Math., Volume 4 (2018), pp. 1-7 | DOI

[24] Kumar, R.; Kumar, R.; Rochon, D. The Fundamental Theorems in the framework of Bicomplex Topological Modules, arXiv, Volume 2011 (2011), pp. 1-14

[25] Kumar, R.; Singh, K. Bicomplex linear operators on bicomplex Hilbert spaces and Littlewood’s subordination theorem, Adv. Appl. Clifford Algebr., Volume 25 (2015), pp. 591-610 | DOI

[26] Kumar, R.; Singh, K.; Saini, H.; Kumar, S. Bicomplex weighted Hardy spaces and bicomplex C∗-algebras, Adv. Appl. Clifford Algebr., Volume 26 (2016), pp. 217-235 | DOI

[27] Lavoie, R. G.; Marchildon, L.; Rochon, D. The bicomplex quantum harmonic oscillator, Nuovo Cimento Soc. Ital. Fis. B, Volume 125 (2010), pp. 1173-1192

[28] Lavoie, R. G.; Marchildon, L.; Rochon, D. Infinite-Dimensional Bicomplex Hilbert Spaces, Ann. Funct. Anal.,, Volume 1 (2010), pp. 75-91 | DOI

[29] Lavoie, R. G.; Marchildon, L.; Rochon, D. Finite-Dimensional Bicomplex Hilbert Spaces, Adv. Appl. Clifford Algebr., Volume 21 (2011), pp. 561-581 | DOI

[30] Luna-Elizarraras, M. E.; Perez-Regalado, C. O.; Shapiro, M. On linear functionals and Hahn-Banach theorems for hyperbolic and bicomplex modules, Adv. Appl. Clifford Algebr., Volume 24 (2014), pp. 1105-1129 | DOI

[31] Luna-Elizarraras, M. E.; Shapiro, M.; Struppa, D. C.; Vajiac, A. Bicomplex Numbers and their Elementary Functions, Cubo, Volume 14 (2012), pp. 61-80 | DOI

[32] Alpay, D.; Luna-Elizarraras, M. E.; Daniele, M. Shapiro; Struppa, C. Basics of Functional Analysis with bicomplex scalars, and bicomplex Schur analysis, Springer International Publishing, New York, 2014 | DOI

[33] Mittag-Leffler, G. Sur la representation analytiqie d’une fonction monogene (cinquieme note), Acta Math., Volume 29 (1905), pp. 101-181

[34] Price, G. B. An Introduction to Multicomplex Spaces and Functions, Marcel Dekker Inc., New York, 1991 | DOI

[35] Riley, J. D. Contributions to the theory of functions of a bicomplex variable, Tohoku Math. J., Volume 5 (1953), pp. 132-165 | DOI

[36] Ringleb, F. Beitrage zur Funktionentheorie in hyperkomplexen Systemen I., Rend. Circ. Mat. Palermo, Volume 57 (1933), pp. 311-340 | DOI

[37] Rochon, D. A Bicomplex Riemann Zeta Function, Tokyo J. Math., Volume 27 (2004), pp. 357-369 | DOI

[38] Rochon, D.; Shapiro, M. On algebraic properties of bicomplex and hyperbolic numbers, An. Univ. Oradea Fasc. Mat., Volume 11 (2004), pp. 71-110

[39] Rochon, D.; Tremblay, S. Bicomplex quantum mechanics. I. The generalized Schrödinger equation, Adv. Appl. Clifford Algebr., Volume 14 (2004), pp. 231-248 | DOI

[40] Rochon, D.; Tremblay, S. Bicomplex quantum mechanics. II. The Hilbert space, Adv. Appl. Clifford Algebr., Volume 16 (2006), pp. 135-157 | DOI

[41] Rönn, S. Bicomplex algebra and function theory, arXiv, Volume 2001 (2001), pp. 1-71

[42] Segre, C. Le rappresentazioni reale delle forme complessee Gli Enti Iperalgebric, Math. Ann., Volume 40 (1892), pp. 413-469 | DOI

[43] Yassen, M. F.; Attiya, A. A.; Agarwal, P. Subordination and Superordination properties for certain family of analytic functions associated with Mittag-Leffler function, Symmetry, Volume 12 (2020), pp. 1-20 | DOI

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