A novel Kind of the multi-index Beta, Gauss, and confluent hypergeometric functions

Volume 23, Issue 2, pp 145--154 http://dx.doi.org/10.22436/jmcs.023.02.07

Publication Date: October 22, 2020 Submission Date: August 01, 2020 Revision Date: September 08, 2020 Accteptance Date: September 18, 2020

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1028 Downloads
• 2435 Views

Authors

Musharraf Ali - Department of Mathematics , G.F. College, Shahjahanpur-242001, India. Mohd Ghayasuddin - Department of Mathematics, Integral University Campus, Shahjahanpur-242001, India. Waseem A. Khan - Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box: 1664, Al Khobar 31952, Saudi Arabia. Kottakkaran Sooppy Nisar - Department of Mathematics, College of Arts and Sciences, Wadi Aldawser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia.

Abstract

This research article elaborates on a novel expansion of the beta function by using the multi-index Mittag-Leffler function. Here, we derive some basic properties of this new beta function and then present a new type of beta dispersal as an application of our proposed beta function. We also introduce a novel expansion of Gauss and confluent hypergeometric functions for our newly initiated beta function. Some important properties of our proposed hypergeometric functions (like integral representations, differential formulae, transformations formulae, summation formulae, and a generating relation) are also pointed out systematically.

Share and Cite

Keywords

• Beta function
• extended beta function
• Gauss hypergeometric function
• confluent hypergeometric function
• multi-index Mittag-Leffler function

MSC

•  33B15
•  33B20
•  33C05
•  33C15
•  33E12

References

• [1] A. A. Al-Gonah, W. K. Mohammed, A new extension of extended gamma and beta functions and their properties, J. Sci. Eng. Res., 9 (2018), 257--270
  • Google Scholar


• [2] M. Ali, M. Ghayasuddin, A note on extended beta, Gauss and confluent hypergeometric functions, Italian J. pure Appl. Math., (), (Accepted)


• [3] M. A. Chaudhry, A. Qadir, M. Rafique, S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math., 78 (1997), 19--32
  • View Article
  • MathSciNet
  • Google Scholar


• [4] M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589--602
  • View Article
  • Google Scholar


• [5] J. Choi, A. K. Rathie, R. K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36 (2014), 357--385
  • View Article
  • MathSciNet
  • Google Scholar


• [6] M. Ghayasuddin, N. U. Khan, M. Ali, A unified presentation of beta, Gauss and confluent hypergeometric functions, (communicated), (2019)


• [7] M. Ghayasuddin, N. U. Khan, M. Ali, A study of extended beta, Gauss and confluent hypergeometric function, Inter. J. Appl. Math., 33 (2020), 1--13
  • View Article


• [8] V. S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math., 118 (2000), 241--259
  • View Article
  • Google Scholar


• [9] V. S. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. Appl., 59 (2010), 1885--1895
  • View Article
  • MathSciNet
  • Google Scholar


• [10] G. M. Mittag-Leffler, Sur la représentation analytique d'une branche uniforme d'une fonction monogène: cinquième note, Acta Math., 29 (1905), 101--182
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] S. Mubeen, G. Rahman, K. S. Nisar, J. Choi, M. Arshad, An extended beta function and its properties, Far East J. Math. Sci., 102 (2017), 1545--1557
  • View Article
  • Google Scholar


• [12] E. Özergin, M. A. Özarslan, A. Altın, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601--4610
  • MathSciNet
  • Google Scholar


• [13] J. Paneva-Konovska, Multi-index ($3m$-parametric) Mittag-Leffler functions and fractional calculus, C. R. Acad. Bulgare Sci., 64 (2011), 1089--1098
  • View Article
  • MathSciNet
  • Google Scholar


• [14] J. Paneva-Konovska, From Bessel to multi-index Mittag-Leffler functions: Enumerable families, series in them and convergence, World Scientific Publishing, London (2016)
  • View Article
  • Google Scholar


• [15] J. Paneva-Konovska, A survey on Bessel type functions as multi-index Mittag-Leffler functions: Differential and integral relations, Int. J. Appl. Math., 32 (2019), 357--380
  • View Article
  • Google Scholar


• [16] R. K. Parmar, A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Matematiche (Catania), 68 (2013), 33--52
  • View Article
  • MathSciNet
  • Google Scholar


• [17] P. I. Pucheta, A new extended beta function, Int. J. Math. Appl., 5 (2017), 255--260
  • View Article
  • Google Scholar


• [18] E. D. Rainville, Special functions, Macmillan Co., New York (1971)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] M. Shadab, S. Jabee, J. Choi, An extended beta function and its application, Far East J. Math. Sci., 103 (2018), 235--251
  • View Article
  • Google Scholar


• [20] H. M. Srivastava, H. L. Manocha, A treatise on generating functions, John Wiley & Sons, New York (1984)
  • MathSciNet
  • Google Scholar
  • MATH


• [21] S. B. Yakubovich, Y. F. Luchko, The hypergeometric approach to integral transform and convolutions, Kluwer Academic Publishers Group, Dordrecht (1994)
  • View Article
  • MathSciNet
  • Google Scholar