---

Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 437--454 | DOI:10.5890/DNC.2023.06.014

$^{1}$ Department of HEAS (Mathematics), Rajasthan Technical University, Kota, India

$^{2}$ Department of Mathematics, Malaviya National Institute of Technology, Jaipur, India

Download Full Text PDF

 

Abstract

Our current research is motivated by the new interesting generalization (Srivastava et al \cite{1}) of a couple of contour-type Mellin-Barnes integral representations of their incomplete $H$-functions $\gamma^{m,\;n}_{p,\;q}(z)$ and $\Gamma^{m,\;n}_{p,\;q}(z)$, and incomplete $\overline{H}$-functions $\overline{\gamma}^{m,\;n}_{p,\;q}(z)$ and $\overline{\Gamma}^{m,\;n}_{p,\;q}(z)$. By virtue of the gamma functions of incomplete type, that is $\gamma(s,x)$ and $ \Gamma(s,x)$, we introduced here a class of the incomplete $I$-functions $^{\gamma}I^{m,\;n}_{p,\;q}(z)$ and $^{\Gamma}I^{m,\;n}_{p,\;q}(z)$ which leads to a natural extension of a class of $I$-functions. The aim of the present insvestigation is to analyze and examine some impressive properties of these incomplete $I$-functions, inclusive of formulas for decomposition, reduction, derivative and several integral transformations, etc. Further, as the application of newly defined functions, we also formulate and solve a generalized fractional kinetic equation in terms of these incomplete $I$-functions. For the corresponding incomplete $\overline{I}$-functions, we demonstrate the simply determinable extensions of the outcome shown here which also hold. We also raising these effects in certain useful specific forms and established results as well.

References

1. [1]	Srivastava, H.M., Saxena, R.K., and Parmar, R.K. (2018), Some families of the incomplete $H$-functions and the incomplete $\overline{H}$-functions and associated integral transforms and operators of fractional calculus with applications, Russian Journal of Mathematical Physics, 25(1), 116-138.

2. [2]	Chaudhry, M.A. and Zubair, S.M. (2002), Extended incomplete gamma functions with applications, Journal of Mathematical Analysis and Applications, 274, 725-745.

3. [3]	Chaudhry, M.A. and Zubair, S.M. (2001), On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall, (CRC Press Company), Boca Raton, London, New York and Washington, D.C.

4. [4]	Chaudhry, M.A. and Qadir, A. (2002), Incomplete exponential and hypergeometric functions with applications to non-central $\chi^2$-distribution, Communications in Statistics- Theory and Methods, 34, 525-535.

5. [5]	\c{C}etinkaya, A. (2013), The incomplete second Appell hypergeometric functions, Applied Mathematics and Computation, 219, 8332-8337.

6. [6]	Choi, J. and Parmar, R.K. (2015), The incomplete Lauricella and fourth Appell functions, Far East Journal of Mathematical Sciences, 96, 315-328.

7. [7]	Choi, J., Parmar, R.K., and Chopra, P. (2016), The incomplete Srivastava's triple hypergeometric functions $\gamma_{B}^{H}$ and $\Gamma_{B}^{H}$, Filomat, 30, 1779-1787.

8. [8]	Choi, J., Parmar, R.K., and Chopra, P. (2014), The incomplete Lauricella and first Appell functions and associated properties, Honam Mathematical Journal, 36, 531-542.

9. [9]	Choi, J. and Parmar, R.K. (2017), The incomplete Srivastava's triple hypergeometric functions $\gamma_{A}^{H}$ and $\Gamma_{A}^{H}$, Miskolc Mathematical Notes, 19(1), 191-200.

10. [10]	Choi, J., Parmar, R.K., and Srivastava, H.M. (2018), The incomplete Lauricella functions of several variables and associated properties and formulas, Kyungpook Mathematical Journal, 58(1), 19-35.

11. [11]	Laadjal, Z., Abdeljawad, T., and Jarad, F.(2020), On existence-uniqueness results for proportional fractional differential equations and incomplete gamma functions, Advances in Difference Equations, 2020, 641.

12. [12]	Mathai, A.M. and Saxena, R.K. (1978), The H-function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd. New Delhi and John Wiley and Sons, Inc. New York.

13. [13]	Fox, C. (1961), The $G$ and $H$-functions as symmetrical Fourier kernels, Transactions of the American Mathematical Society, 98, 395-429.

14. [14]	Mathai, A.M., Saxena, R.K., and Haubold, H.J. (2010), The $H$-Functions: Theory and Applications, Springer, New York.

15. [15]	Srivastava, H.M., Chaudhry, M.A., and Agarwal, R.P. (2012), The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms and Special Functions, 23, 659-683.

16. [16]	Srivastava, R. (2013), Some properties of a family of incomplete hypergeometric functions, Russian Journal of Mathematical Physics, 20, 121-128.

17. [17]	Srivastava, R., Agarwal, R., and Jain, S. (2017), A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas, Filomat, 31, 125-140.

18. [18]	Purohit, S.D., Khan, A.M., Suthar, D.L., and Dave, S. (2021), The impact on raise of environmental pollution and occurrence in biological populations pertaining to incomplete $H$-function, National Academy Science Letters, 44(3), 243-254.

19. [19]	Meena, S., Bhatter, S., Jangid, K., and Purohit, S.D. (2020), Certain integral transforms concerning the product of family of polynomials and generalized incomplete functions, Moroccan Journal of Pure and Applied Analysis (MJPAA), 6(2), 263-266.

20. [20]	Meena, S., Bhatter, S., Jangid, K., and Purohit, S.D. (2020), Some expansion formulas for incomplete $H$ and $\overline{H}$-functions involving Bessel functions, Advances in Difference Equations, 2020, 562.

21. [21]	Jangid, N.K., Joshi, S., Purohit, S.D., and Suthar, D.L. (2021), Fractional derivatives and expansion formulae of incomplete $H$ and $\overline{H}$-functions, Advances in the Theory of Nonlinear Analysis and its Application, 5(2), 193-202.

22. [22]	Rathie, A.K. (1997), A new generalization of generalized Hypergeometric functions, Le Matematiche, LII, 297-310.

23. [23]	Oberhettinger, F. (1974), Tables of Mellin Transforms, Springer, New York.

24. [24]	Lavoie, J.L. and Trottier, G. (1969), On the sum of certain Appell's series, Ganita, 20, 31-32.

25. [25]	Qureshi, M.I., Quraishi, K.A., and Pal, R. (2011), Some definite integrals of Gradshtenyn Ryzhil and other integrals, Global Journal of Science Frontier Research, 11(4), 75-80.

26. [26]	MacRobert, T.M. (1961), Beta functions formulas and integrals involving E-function, Mathematische Annalen, 142, 450-452.

27. [27]	Suthar, D.L., Purohit, S.D., and Araci, S. (2020), Solution of fractional kinetic equations associated with the $(p, q)$-Mathieu-type series, Discrete Dynamics in Nature and Society, 2020, 8645161.

28. [28]	Habenom, H., Suthar, D.L., Baleanu, D., and Purohit, S.D. (2021), A numerical simulation on the effect of vaccination and treatments for the fractional hepatitis B model, Journal of Computational and Nonlinear Dynamics, 16(1), 011004.

29. [29]	Baleanu, D., Jajarmi, A., Mohammadi, H., and Rezapour, S. (2020), A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Solitons Fractals, 134, 109705.

30. [30]	Baleanu, D., Mohammadi, H., and Rezapour, S. (2020), A fractional differential equation model for the COVID-19 transmission by using Caputo-Fabrizio derivative. Advances in Difference Equations, 2020, 299.

31. [31]	Haubold, H.J. and Mathai, A.M. (2000), The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science, 327, 53-63.

32. [32]	Saxena, R.K. and Kalla, S.L. (2008), On the solutions of certain fractional kinetic equations, Applied Mathematics and Computation, 199, 504-511.