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Research Article

The incomplete matrix beta function and its application to first Appell hypergeometric matrix function

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Pages 2003-2021 | Received 17 Apr 2022, Accepted 01 Jun 2023, Published online: 18 Jun 2023

References

  • Gradshteyn I, Ryzhik I. Tables of integrals, series, and products. San Diego: Academic Press; 2000.
  • Dutka J. The incomplete beta function—a historical profile. Arch Hist Exact Sci. 1981;24:11–29. doi: 10.1007/BF00327713
  • Jack Ng Y, Van Dam H. Neutrix calculus and finite quantum field theory. J Phys A. 2005;38:317–323. doi: 10.1088/0305-4470/38/18/L03
  • Jack Ng Y, Van Dam H. An application of neutrix calculus to quantum field theory. Internat J Modern Phys A. 2006;21:297–312. doi: 10.1142/S0217751X06025225
  • Li A, Qin H. Some transformation properties of the incomplete beta function and its partial derivative. IAENG Int J Appl Math. 2019;49:1–14.
  • Lin M, Fisher B, Orankitjaroen S. Some results on the beta function and the incomplete beta function. Asian-Eur J Math. 2015;8:1–11. doi: 10.1142/S1793557115500485
  • Ozçaḡ E, Ege Ī, Gçay H. An extension of the incomplete beta function for negative integers. J Math Anal Appl. 2008;338:984–992. doi: 10.1016/j.jmaa.2007.05.075
  • Ozçaḡ E, Ege Ī, Gçay H, et al. An partial derivatives of the incomplete beta function. Appl Math Lett. 2008;21:675–681. doi: 10.1016/j.aml.2007.07.020
  • Sun Z, Qin H, Li A. Extension of the partial derivatives of the incomplete beta function for complex values. Appl Math Comput. 2016;275:63–71. doi: 10.1016/j.amc.2015.11.054
  • Van der Corput J. Introduction to the neutrix calculus. J Anal Math. 1959–1960;7:281–398. doi: 10.1007/BF02787689
  • Akel M, Bakhet A, Abdalla M, et al. On degenerate gamma matrix functions and related functions. Linear Multilinear Algebra. 2022. doi: 10.1080/03081087.2022.2040942
  • Bakhet A, He F, Yu M. On the matrix version of extended Bessel functions and its application to matrix differential equations. Linear Multilinear Algebra. 2022;70:5661–5680. doi: 10.1080/03081087.2021.1923629
  • Karp DB. Normalized incomplete beta function: log-concavity in parameters and other properties. J Math Sci. 2016;217:91–107. doi: 10.1007/s10958-016-2958-z
  • Leroya C, Ishkhanyanb A. Expansions of the solutions of the confluent Heun equation in terms of the incomplete beta and the Appell generalized hypergeometric functions. Integral Transforms Spec Funct. 2015;26:451–459. doi: 10.1080/10652469.2015.1019490
  • Nemes G, Daalhuis A. Uniform asymptotic expansion for the incomplete beta function. SIGMA. 2016;12:101–106. doi: 10.3842/SIGMA.2016.101
  • Ojo M. A new approximation to the incomplete beta function. Commun Stat Theory Methods. 1988;17:1423–1435. doi: 10.1080/03610928808829689
  • Temme N. Asymptotic inversion of the incomplete beta function. J Comput Appl Math. 1992;41:145–157. doi: 10.1016/0377-0427(92)90244-R
  • Srivastava H, Chaudhry M, Agarwal R. The incomplete Pochhammer symbols and their applications to hypergeometric and related functions. Integral Transforms Spec Funct. 2012;23:659–683. doi: 10.1080/10652469.2011.623350
  • Cetinkaya A. The incomplete second Appell hypergeometric functions. Appl Math Comput. 2013;219:8332–8337. https://doi.org/10.1016/j.amc.2012.11.050
  • Srivastava R. Some properties of a family of incomplete hypergeometric functions. Russ J Math Phys. 2013;20:121–128. doi: 10.1134/S1061920813010111
  • Choi J, Agarwal P. Certain class of generating functions for the incomplete hypergeometric functions. Abstr Appl Anal. 2014;2014:Article ID 714560, 348–352. doi: 10.1155/2014/714560
  • Bansal M, Kumar D, Nisar K, et al. Application of incomplete H-functions in determination of Lambert's law. J Interdiscip Math. 2019;22:1205–1212. doi: 10.1080/09720502.2019.1709319
  • Bansal M, Kumar D, Singh J, et al. On the solutions of a class of integral equations pertaining to incomplete H-function and incomplete H-function. Mathematics. 2020;8:819. doi: 10.3390/math8050819
  • Parmar R. Certain properties of extended complete and incomplete beta functions. AIP Conf Proc. 2016;1728. Article ID 020695. doi: 10.1063/1.4946746
  • Özarslan M, Ustaoglu C. Incomplete Caputo fractional derivative operators. Adv Differ Equ. 2018;2018:209. doi: 10.1186/s13662-018-1656-1
  • Özarslan M, Ustaoǧlu C. Some incomplete hypergeometric functions and incomplete Riemann–Liouville fractional integral operators. Mathematics. 2019;7:483, 1–18. doi: 10.3390/math7050483
  • Srivastava H, Agarwal P. Certain fractional integral operators and the generalized incomplete hypergeometric functions. Appl Appl Math. 2013;8:333–345. http://www.pvamu.edu/mathematics/wp-content/uploads/sites/49/01_r631-srivastava-2013.pdf.
  • Zou C, Yu M, Bakhet A, et al. On the matrix versions of incomplete extended gamma and beta functions and their applications for the incomplete Bessel matrix functions. Complexity. 2021;2021:Article ID 5586021. doi: 10.1155/2021/5586021
  • Abdalla M. Special matrix functions: characteristics, achievements and future directions. Linear Multilinear Algebra. 2020;68:1–28. doi: 10.1080/03081087.2018.1497585
  • Jódar L, Cortés JC. Some properties of gamma and beta matrix functions. Appl Math Lett. 1998;11:89–93. doi: 10.1016/S0893-9659(97)00139-0
  • Jódar L, Cortés JC. On the hypergeometric matrix function. J Comp Appl Math. 1998;99:205–217. doi: 10.1016/S0377-0427(98)00158-7
  • Nagara D, Correa A, Gupta A. Extended matrix variate gamma and beta functions. J Multivar Anal. 2013;122:53–69. doi: 10.1016/j.jmva.2013.07.001
  • Salem A. On a q-gamma and a q-beta matrix functions. Linear Multilinear Algebra. 2012;60:683–696. doi: 10.1080/03081087.2011.627562
  • Sastre J, Jódar L. Asymptotics of the modified Bessel and incomplete gamma matrix functions. Appl Math Lett. 2003;16(6):815–820. doi: 10.1016/S0893-9659(03)90001-2
  • Abdalla M. On the incomplete hypergeometric matrix functions. Ramanujan J. 2017;43(3):663–678. doi: 10.1007/s11139-016-9795-z
  • Bakhet A, Jiao Y, He F. On the Wright hypergeometric matrix functions and their fractional calculus. Integral Transforms Spec Funct. 2019;30:138–156. doi: 10.1080/10652469.2018.1543669
  • Verma A, Yadav S. On the incomplete second Appell hypergeometric matrix functions. Linear Multilinear Algebra. 2021;69:1747–1760. doi: 10.1080/03081087.2019.1640178
  • Verma A. On the incomplete Srivastava's triple hypergeometric matrix functions. Quaest Math. 2021;44:881–904. doi: 10.2989/16073606.2020.1753123
  • Abd-Elmageed H, Abdalla M, Abul-Ez M, et al. Some results on the first Appell matrix function F1(A;B,B′;C;z,w). Linear Multilinear Algebra. 2020;68:278–292. doi: 10.1080/03081087.2018.1502254
  • Hidan H, Bakhet A, Abd-Elmageed H, et al. Matrix-valued hypergeometric Appell-type polynomials. Electron Res Arch. 2022;30:2964–2980. doi: 10.3934/era.2022150

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