Certain results concerning (p, q)-parameterized beta logarithmic function and their properties

Abstract

The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed.

Keywords

• Beta function
• logarithmic mean
• beta distribution
• hypergeometric function
• confluent hypergeometric function

References

1. Andrews, G.E., Askey, R., Roy, R., Special Functions, Cambridge University Press, Cambridge, 1999.
2. Chaudhry, M.A., Zubair, S.M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55(1) (1994), 99–123. https://doi.org/10.1016/0377-0427(94)90187-2
3. Chaudhry, M.A., Qadir, A., Rafique, M., Zubair, S.M., Extension of Euler’s beta function, J. Comput. Appl. Math., 78 (1997), 19–32. https://doi.org/10.1016/S0377-0427(96)00102-1
4. Chaudhry, M.A., Qadir, A., Srivastava, H.M., Paris, R.B., Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589–602. https://doi.org/10.1016/j.amc.2003.09.017
5. Choi, J., Rathie, A.K., Parmar, R.K., Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Mathematical J., 36(2) (2014), 357–385. https://doi.org/ 10.5831/hmj.2014.36.2.357
6. Kaba, D.G., On some inequalities satisfied by beta and gamma functions, South African Statistical Journal, 12(1) (1978), 25–31.
7. Khan, N.U., Aman, M., Usman, T., Extended beta, hypergeometric and confluent hypergeometric functions via multi-index Mittag-Leffler function, Proc. of the Jangjeon Mathematical Society, 25(1) (2022), 43–58. https://doi.org/ 10.17777/pjms2022.25.1.43
8. Khan, N.U., Khan, M. I., Khan, O., Evaluation of transforms and fractional calculus of new extended Wright function, Int. J. Appl. Comput. Math., 8(4) (2022), 163. https://doi.org/10.1007/s40819-022-01365-7

9. Khan, N.U., Husain, S., A note on extended beta function involving generalized Mittag-Leffler function and its applications, TWMS J. of Appl. and Eng. Math., 12(1) (2022), 71–81.
10. Kumar, P., Singh, S.P., Dragomir, S.S., Some inequalities involving beta and gamma functions, Nonlinear Analysis Forum, 6(1) (2001), 143–150.
11. Luke, Y.L., The Special Functions and Their Approximations, Academic Press, New York, 1969.
12. Özergin, E., Özarslan, M.A., Altin, A., Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235(16) (2011), 4601–4610. https://doi.org/10.1016/j.cam.2010.04.019
13. Rainville, E.D., Special Functions, The Macmillan Co. Inc., New York, 1960.
14. Raissouli, M., Chergui, M., On a new parameterized beta function, Proc. of the Institute of Math. and Mechanics, National Academy of Sciences of Azerbaijan, 48(1) (2022), 132–139. https://doi.org/10.30546/2409-4994.48.1.2022.132
15. Saif, M., Khan, F., Nisar, K.S., Araci, S., Modified Laplace transform and its properties, J. Math. Computer Sci., 21(2) (2020), 127–135. https://doi.org/ 10.22436/jmcs.021.02.04