On the extended Wright hypergeometric matrix function and its properties

Year 2023, Volume: 72 Issue: 3, 606 - 617, 30.09.2023

Halil Gezer , Cem Kaanoglu

https://doi.org/10.31801/cfsuasmas.1147745

Abstract

Recently, Bakhet et al. [9] presented the Wright hypergeometric matrix function $_{2}R_{1}^{(\tau )}(A,B;C;z)$ and derived several properties. Abdalla [6] has since applied fractional operators to this function. In this paper, with the help of the generalized Pochhammer matrix symbol $(A;B)_{n}$ and the generalized beta matrix function $\mathcal{B}(P,Q;\mathbb{X})$, we introduce and study an extended form of the Wright hypergeometric matrix function, $_{2}R_{1}^{(\tau )}((A,\mathbb{A}),B;C;z;\mathbb{X}).$ We establish several potentially useful results for this extended form, such as integral representations and fractional derivatives. We also derive some properties of the corresponding incomplete extended Wright hypergeometric matrix function.

Keywords

Wright hypergeometric matrix function, generalized hypergeometric functions, Riemann-Liouville fractional derivative

References

• Abd-Elmageed, H., Hidan, M., Abdalla, M., Investigation for the k-analogue of $\tau$-Gauss hypergeometric matrix functions and associated fractional calculus, Linear and Multilinear Algebra, (2022), 1-14. https://doi.org/10.1080/03081087.2022.2161459
• Abdalla, M., On the incomplete hypergeometric matrix functions, Ramanujan J., 43 (2017), 663-678. https://doi.org/10.1007/s11139-016-9795-z
• Abdalla, A., Akel, M., Contribution of using Hadamard fractional integral operator via Mellin integral transform for solving certain fractional kinetic matrix equations, Fractal and Fractional, 6(6) (2022), 305. https://doi.org/10.3390/ fractalfract6060305
• Abdalla, M., Bakhet, A., Extended Gauss hypergeometric matrix functions, Iran J Sci Technol Trans Sci., 42 (2018), 1465-1470. https://doi.org/10.1007/s40995-017-0183-3
• Abdalla, M., Bakhet, A., Extension of beta matrix function, Asian J Math Comput Res., 9 (2016), 253-264.
• Abdalla, M., Fractional operators for the Wright hypergeometric matrix functions, Advances in Difference Equations, (2020), 246. https://doi.org/10.1186/s13662-020-02704-y
• Abul-Dahab, M. A., Bakhet, A. K., A certain generalized gamma matrix functions and their properties, J. Ana. Num. Theor., 3(1) (2015), 63-68. https://dx.doi.org/10.12785/jant/030110
• Bakhet, A., Hyder, A. A., Almoneef, A. A., Niyaz, M., Soliman, A. H., On new matrix version extension of the incomplete Wright hypergeometric functions and their fractional calculus, Mathematics, 10(22) (2022), 4371. https://doi.org/10.3390/math10224371
• Bakhet, A., Jiao, Y., He, F., On the Wright hypergeomertric matrix functions and their fractional calculus, Integral Transforms Spec. Funct., 30 (2019), 138-156. https://doi.org/10.1080/10652469.2018.1543669
• Dwivedi, R., Sanjhira, R., On the matrix function $_{p}R_{q}(A;B;z)$ and its fractional calculus properties, Communications in Mathematics, 31(1) (2023), 43-56. https://doi.org/10.46298/cm.10205
• Hidan, M., Akel, M., Abd-Elmageed, H., Abdalla, M., Solution of fractional kinetic equations involving extended $(k,\tau)$-Gauss hypergeometric matrix functions, AIMS Math., 7(8) (2022), 14474-14491. https://doi.org/10.3934/math.2022798
• Jodar, L., Cortes, J. C., Some properties of gamma and beta matrix functions, Appl. Math. Lett., 11 (1998), 89-93. https://doi.org/10.1016/S0893-9659(97)00139-0
• Jodar, L., Cortes, J. C., On the hypergeometric matrix functions, J. Compute. Appl. Math., 99 (1998), 205-217. https://doi.org/10.1016/S0377-0427(98)00158-7
• Khammash, G. S., Agarwal, P., Choi, J., Extended k-gamma and k-beta functions of matrix arguments, Mathematics, 8 (2020), 1715. https://doi.org/10.3390/math8101715
• Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Elsevier (North-Holland) Science Publishers, Amsterdam, (2006), 204.
• Özarslan, M. A., Ustaoğlu, C., Incomplete Caputo fractional derivative operators, Adv. Differ. Equ., (2018), 209. https://doi.org/10.1186/s13662-018-1656-1
• Özarslan, M. A., Ustaoğlu, C., Some incomplete hypergeometric functions and incomplete Riemann-Liouville fractional integral operators, Mathematics, 7 (2018), 483. https://doi.org/10.3390/math7050483
• Verma, A., On the incomplete Srivastava‘s triple hypergeometric matrix functions, Quaest Math., (2020), 1-24. https://doi.org/10.2989/16073606.2020.1753123
• Verma, A., Yadav, S., On the incomplete second Appell hypergeometric matrix functions, Linear Multilinear Algebra, (2019). https://doi.org/10.1080/03081087.2019.1640178
• Verma, A., Dwivedi, R., Sahai, V., Some extended hypergeometric matrix functions and their fractional calculus, (2020), arXiv:2011.00772v1. https://doi.org/10.48550/arXiv.2011.00772
• Zou, C., Yu, M., Bakhet, A., He, F., On the matrix versions of incomplete extended gamma and beta functions and their applications for the incomplete Bessel, Complexity, (2020). https://doi.org/10.1155/2021/5586021