Some Generalized Special Functions and their Properties

Abstract

In this present paper, first, we investigate a new generalized Pochhammer's symbol and its various properties in terms of a new symbol $(s; k)$, where $s; k > 0$. Then, we define a new generalization of gamma and beta functions and their various associated properties in the form of $(s; k)$. Also, we define a new generalization of hypergeometric functions and develop differential equations for generalized hypergeometric functions in the form of $(s; k)$. We present that generalized hypergeometric functions are the solution of the said differential equation. Furthermore, some useful results and properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

Keywords

• Pochhammer symbol
• beta function
• Gamma function
• generalized Pochhammer symbol
• generalized hypergeometric function

Kaynakça

1. [1] S. Araci, G. Rahman, A. Ghaffar, K.S. Nisar, Fractional calculus of extended Mittag-Lefler function and its applications to statistical distribution, Math., 7(3) (2019), 248.
2. [2] M.A. Chaudhry, S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comp. Appl. Math., 55(1) (1994), 99-123.
3. [3] M.A. Chaudhry, S.M. Zubair, On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, J. Comp. Appl. Math., 59(3) (1995), 253-284.
4. [4] M.A. Chaudhry, S.M. Zubair, Extended incomplete gamma functions with applications, J. Math. Anal. Appl., 274(2) (2002), 725-745.
5. [5] P. Agarwal, Q. Al-Mdallal, Y.J. Cho, S. Jain, Fractional differential equations for the generalized Mittag-Leffler function. Adv. Di?. Eq. (2018)(1), 1-8.
6. [6] Q. Al-Mdallal, M. Al-Refai, M. Syam, M.D.K. Al-Srihin, Theoretical and computational perspectives on the eigenvalues of fourth-order fractional Sturm-Liouville problem, Int. J. Comp. Math., 95(8) (2018), 1548-1564.
7. [7] A. Babakhani, Q. Al-Mdallal, On the existence of positive solutions for a non-autonomous fractional differential equation with integral boundary conditions, Comp. Meth. Diff. Eq., 9(1) (2021), 36-51.
8. [8] F. Jarad, T. Abdeljawad, A modi?ed Laplace transform for certain generalized fractional operators, Nonlin. Anal., 1(2) (2018), 88-98.

9. [9] R. Diaz, C. Teruel, (q,k)-Generalized gamma and beta functions, J. Nonlin. Math. Phy., 12(2005), 118-134.
10. [10] R. Diaz, E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divul. Math., 15(2007), 179-192.
11. [11] R. Diaz, C. Ortiz, E. Pariguan, On the k-gamma q-distribution, Cent. Eur. J. Math., 8 (2010), 448-458.
12. [12] Kokologiannaki, CG: Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sci., 5, 653-660 (2010).
13. [13] V. Krasniqi, A limit for the k-gamma and k-beta function. Int. Math. Forum. 5 (2010), 1613-1617.
14. [14] M. Mansour, Determining the k-generalized gamma function Γ k (x) by functional equations. International Journal of Con- temporary Mathematical Sciences. 4, 1037-1042 (2009).
15. [15] F. Merovci, Power product inequalities for the Γ k function. Int. J. of Math. Analysis., 4(2010), 1007-1012.
16. [16] S. Mubeen, S. Iqbal, Gruss type integral inequalities for generalized Riemann-Liouville k-fractional integrals, J. Ineq. Appl. 109 (2016), pp.13. Available online at https://doi.org/10.1186/s13660-016-1052-x.
17. [17] S. Mubeen, S. Iqbal, Z. Iqbal. On Östrowski type inequalities for generalized k-fractional integrals, J. Ineq. Spec. Funct., 8 (2017), no. 3, 107-118.
18. [18] P. Agarwal, M. Jleli, M. Tomar. Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Ineq. Appl., 2017(55), 10. Available online at https://doi.org/10.1186/s13660-017-1318-y.
19. [19] C.-J. Huang, G. Rahman, K. S. Nisar, A. Ghaffar, and F. Qi, Some inequalities of the Hermite-Hadamard type for k-fractional conformable integrals, Austral. J. Math. Anal. Appl., 16(1) (2019).
20. [20] G. Rahman, K.S. Nisar, A. Ghaffar, F. Qi, Some inequalities of the Gruss type for conformable k-fractional integral operators, RACSAM, (2020) 114: 9. https://doi.org/10.1007/s13398-019-00731-3.
21. [21] G.Farid, G.M. Habullah, An extension of Hadamard fractional integral. Int. J. Math. Anal., 9(10), 471-482 (2015).
22. [22] M. Tomar, S. Mubeen, J. Choi, Certain inequalities associated with Hadamard k-fractional integral operators, J. Ineq. Appl., (2016) (234), (2016).
23. [23] G. Farid, A.U. Rehman, M. Zahra, on Hadamard-type inequalities for k-fractional integrals, Kon. J. Math., 4(2), 79-86 (2016).
24. [24] S. Iqbal, S. Mubeen, M. Tomar, On Hadamard k-fractional integrals, J. Fract. Cal. Appl., 9(2) (2018), 255-267.
25. [25] S. Habib, S. Mubeen, M.N. Naeem, F. Qi, Generalized k-fractional conformable integrals and related inequalities, AIMS Math., 4(3), 343-358.
26. [26] M. Samraiz, E. Set, M. Hasnain, G. Rahman, On an extension of Hadamard fractional derivative, J. Ineq. Appl. (2019) 2019:263. https://doi.org/10.1186/s13660-019-2218-0
27. [27] E. Set, M.A. Noor, M.U. Awan, A. G¨ozpinar, Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl. (2017), 169, 10. Available online at https://doi.org/10.1186/s13660-017-1444-6.
28. [28] K.S. Nisar, G. Rahman, J. Choi, Certain Gronwall type inequalities associated with riemann-liouville and hadamard k-fractional derivatives and their applications, E. Asi. Math. J., 34 (2018), 249-263.
29. [29] G. Rahman, S. Mubeen, K.S. Nisar, On generalized k-fractional derivative operator, AIMS Mathematics, 5(3), 2019, 1936-1945.
30. [30] K. Jangid, S.D. Prohit, K.S. Nisar, T. Abdeljawad, Certain Generalized Fractional Integral Inequalities. Advances in the Theory of Nonlinear Analysis and its Application, 4(4) (2020), 252-259.
31. [31] F. Qi, G. Rahman, S.M. Hussain, Some inequalities of Chebysev Type for conformable k-Fractional integral operators, Symmetry, 10 (2018), 614.
32. [32] S. Mubeen, Solution of Some integral equations involving confluent k-hypergeometric functions, Appl. Math. 4, 9-11 (2013).
33. [33] S. Mubeen, G.M. Habibullah, An integral representation of some k-hypergeometric functions, Int. Math. Forum. J. Th. Appl., vol.7, 1-4. 203-207, 2012.
34. [34] S. Mubeen, Solution of some integral equations involving confluent k-hypergeometric functions, Applied Mathematics, vol. 4, no. 7A, pp. 9-11, 2013.
35. [35] S. Mubeen, A. Rehman, A Note on k-Gamma function and Pochhammer k-symbol, J. Inf. Math. Sci. 2014, 6, 93-107.
36. [36] S. Mubeen, M. Naz, A. Rehman, G. Rahman, Solutions of k-hypergeometric differential equations, J. Appl. Math. 2014, (2014), 1-13.
37. [37] S. Li, Y. Dong, k-hypergeometric series solutions to One type of non-homogeneous k-hypergeometric equations, Symmetry (2019), 11, 262; doi:10.3390/sym11020262
38. [38] G. Rahman, M. Arshad, S. Mubeen, Some results on generalized hypergeometric k-functions, Bull. Math. Anal. Appl. 8(3) (2016), 66-77.
39. [39] S. Mubeen, C.G. Kokologiannaki, G. Rahman, M. Arshad, Z. Iqbal, Properties of generalized hypergeometric k-functions via k-fractional calculus, Far East Journal of Applied Mathematics, 96(6) (2017), 351-372.
40. [40] F. Qi, A. Wand Geometric interpretations and reversed versions of Young's integral inequality, Adv. Theory Nonlin. Analy. Appl.. 5 (2021) No.1, 1-6.
41. [41] N. Adjimi, M. Benbachir, Katugampola fractional differential equation with Erdelyi-Kober integral boundary conditions, Adv. Theory of Nonlin. Anal. Appl., 5 (2021) No. 2, 215-228.