Abstract

In the present research note, we define a new extension of beta function by making use of the multi-index Mittag-Leffler function. Here, first we derive its fundamental properties and then we present a new type of beta distribution as an application of our proposed beta function. Moreover, we present a new extension of Gauss and confluent hypergeometric functions in terms of our newly introduced beta function. Some interesting properties of our extended hypergeometric functions (like integral representations, differential formulae, transformations and summation formulae and a generating relation) are also indicated in the last section.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 1
Year: 2020

DOI: 10.12732/ijam.v33i1.1

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References

1. [1] J. Choi, A.K. Rathie, R.K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36, No 2 (2014), 357-385.
2. [2] M.A. Chaudhry, A. Qadir, M. Rafique and S.M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math., 78, No 1 (1997), 19-32.
3. [3] M.A. Chaudhry, A. Qadir, H.M. Srivastava and R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159, No 2 (2004), 589-602.
4. [4] V. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. with Appl., 59, No 5 (2010), 1885-1895.
5. [5] V.S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math., 118, No 1-2 (2000), 241-259.
6. [6] G.M. Mittag-Leffler, Sur la representation analytique d’une branche uniforme d’une function monogene, Acta Math., 29 (1905), 101-182.
7. [7] E. Özergin, M.A. Özarslan, A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610.
8. [8] J. Paneva-Konovska, A survey on Bessel type functions as multi-index Mittag-Leffler functions: Differential and integral relations, Intern. J. Appl. Math., 32, No 3 (2019), 357-380; doi: http://dx.doi.org/10.12732/ijam.v32i3.1.
9. [9] R.K. Parmar, A new generalization of Gamma, Beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, LXVIII (2013), 33-52.
10. [10] E.D. Rainville,Special Functions, Macmillan Co., New York (1960); Repr. by Chelsea Publ. Co., Bronx, New York (1971).
11. [11] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Ltd., Chichester), J. Wiley and Sons, N. York-Chichester-Brisbane-Toronto (1984).
12. [12] M. Shadab, S. Jabee and J. Choi, An extension of beta function and its application, Far East J. Math. Sci. 103, No 1 (2018), 235-251.