Certain Results on Extended Beta and Related Functions Using Matrix Arguments

Year 2024, Issue: 49, 16 - 29, 31.12.2024

Nabiullah Khan , Rakibul Sk , Saddam Husain

https://doi.org/10.53570/jnt.1534850

Abstract

In this study, we present and explore extended beta matrix functions (EBMFs) and their key properties. By utilizing the beta matrix function (BMF), we introduce novel extensions of the Gauss hypergeometric matrix function (GHMF) and Kummer hypergeometric matrix function (KHMF). We delve into their integral representations, recurrence relations, transformation properties, and differential formulas. Additionally, we investigate their statistical applications, mainly focusing on the beta distribution, and derive expressions for the mean, variance, and moment-generating functions. Furthermore, we apply EBMFs to develop the Appell matrix function (AMF) and Lauricella matrix function (LMF) and their integral forms.

Keywords

Beta matrix function, Gauss and Kummer hypergeometric matrix functions, Appell and Lauricella matrix functions

References

• M. Abdalla, A. Bakhet, Extended Gauss hypergeometric matrix functions, Iranian Journal of Science and Technology, Transactions A: Science 42 (2018) 1465-1470.
• M. Abdalla, A. Bakhet, Extension of beta matrix function, Asian Journal of Mathematics and Computer Research 9 (3) (2016) 253-264.
• M. Abul-Dahab, A. Bakhet, A certain generalized gamma matrix functions and their properties, Journal of Analysis and Number Theory 3 (1) (2015) 63-68.
• A. Bakhet, Y. Jiao, F. He, On the Wright hypergeometric matrix functions and their fractional calculus, Integral Transforms and Special Functions 30 (2) (2019) 138-156.
• B. Çekim, New kinds of matrix polynomials, Miskolc Mathematical Notes 14 (3) (2013) 817-826.
• B. Çekim, Generalized Euler's beta matrix and related functions, in: T. E. Simos, G. Psihoyios, Ch. Tsitouras (Eds.), 11th International Conference of Numerical Analysis and Applied Mathematics, Rhodes, 2013, pp. 1132-1135.
• R. Dwivedi, V. Sahai, A note on the Appell matrix functions, Quaestiones Mathematicae 43 (3) (2020) 321-334.
• R. Goyal, P. Agarwal, G. I. Oros, S. Jain, Extended beta and gamma matrix functions via 2-parameter Mittag-Leffler matrix function, Mathematics 10 (6) (2022) 892 8 pages.
• M. Izadi, H. M. Srivastava, A novel matrix technique for multi-order pantograph differential equations of fractional order, Proceedings of the Royal Society A 477 (2253) (2021) 20210321 21 pages.
• S. Jain, R. Goyal, G. I. Oros, P. Agarwal, S. Momani, A study of generalized hypergeometric matrix functions via two-parameter Mittag–Leffler matrix function, Open Physics 20 (1) (2022) 730-739.
• L. Jodar, J. C. Cortés, Some properties of gamma and beta matrix functions, Applied Mathematics Letters 11 (1) (1998) 89-93.
• L. Jodar, J. C. Cortés, On the hypergeometric matrix function, Journal of Computational and Applied Mathematics 99 (1-2) (1998) 205-217.
• G. S. Khammash, P. Agarwal, J. Choi, Extended k-Gamma and k-Beta functions of matrix arguments, Mathematics, 8 (10) (2020) 1715 13 pages.
• A. Verma, R. Dwivedi, V. Sahai, Some extended hypergeometric matrix functions and their fractional calculus, Mathematics in Engineering, Science and Aerospace 13 (4) (2022) 1131-1140.
• N. U. Khan, S. Husain, A novel beta matrix function via Wiman matrix function and their applications, Analysis 43 (4) (2023) 255-266.
• R. Dwivedi, V. Sahai, On the hypergeometric matrix functions of several variables, Journal of Mathematical Physics 59 (2) (2018) 023505 15 pages.
• A. Verma, S. Bajpai, K. S. Yadav, Some results of new extended beta, hypergeometric, Appell and Lauricella matrix functions, Research in Mathematics 9 (1) (2022) 2151555 9 pages.
• G. H. Golub, C. F. Van Loan, Matrix computations, 4th Edition, Johns Hopkins University Press, Baltimore, 2013.
• G. B. Folland, Fourier analysis and its applications, American Mathematical Society, Providence, 2009.
• J. Greene, Hypergeometric functions over finite fields, Transactions of the American Mathematical Society 301 (1) (1987) 77-101.
• R. Dwivedi, V. Sahai, On the hypergeometric matrix functions of two variables, Linear and Multilinear Algebra 66 (9) (2018) 1819-1837.
• H. M. Srivastava, H. L. Manocha, A treatise on generating functions, John Wily and Sons, New York 1984.