Certain Results on Extended Beta and Related Functions Using Matrix Arguments
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.