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Year 2022, Volume: 43 Issue: 4, 684 - 695, 27.12.2022

Enes Ata , İ. Onur Kıymaz

https://doi.org/10.17776/csj.1005486
Cited By: 10

Abstract

References

  • [1] Chaudhry M.A., Zubair S.M., Generalized Incomplete Gamma Functions with Applications, J.Comput. Appl. Math., 55 (1) (1994) 99-124.
  • [2] Chaudhry M.A., Qadir A., Rafique M., Zubair S.M., Extension of Euler’s Beta Function, J. Comput. Appl. Math., 78 (1) (1997) 19-32.
  • [3] Chaudhry M.A., Qadir A., Srivastava H.M., Paris R.B., Extended Hypergeometric and Confluent Hypergeometric Functions, Appl. Math. Comput., 159 (2) (2004) 589-602.
  • [4] Ata E., Kıymaz İ.O., A Study on Certain Properties of Generalized Special Functions Defined by Fox-Wright Function, Appl. Math. Nonlinear Sci., 5 (1) (2020) 147-162.
  • [5] Özergin E., Özarslan M.A., Altın A., Extension of Gamma, Beta and Hypergeometric Functions, J. Comput. Appl. Math., 235 (16) (2011) 4601-4610.
  • [6] Lee D.M., Rathie A.K., Parmar R.K., Kim Y.S., Generalization of Extended Beta Function, Hypergeometric and Confluent Hypergeometric Functions, Honam Math. J., 33 (2) (2011) 187-206.
  • [7] Parmar R.K., A New Generalization of Gamma, Beta, Hypergeometric and Confluent Hypergeometric Functions, Le Matematiche, 68 (2) (2013) 33-52.
  • [8] Srivastava H.M., Agarwal P., Jain S., Generating Functions for the Generalized Gauss Hypergeometric Functions, Appl. Math. Comput., 247 (C) (2014) 348-352.
  • [9] Shadab M., Saime J., Choi J., An Extended Beta Function and Its Applications, J. Math. Sci., 103 (1) (2018) 235-251.
  • [10] Rahman G., Mubeen S., Nisar K.S., A New Generalization of Extended Beta and Hypergeometric Functions, J. Frac. Calc. Appl., 11 (2) (2020) 32-44.
  • [11] Çetinkaya A., Kıymaz İ.O., Agarwal P., Agarwal R.A., A Comparative Study on Generating Relations for Generalized Hypergeometric Functions via Generalized Fractional Operators, Adv. Differ. Equ., 2018 (1) (2018) 1-11.
  • [12] Goswami A., Jain S., Agarwal P., Aracı S., A Note on the New Extended Beta and Gauss Hypergeometric Functions, Appl. Math. Infor. Sci., 12 (1) (2018) 139-144.
  • [13] Choi J., Rathie A.K., Parmar R.K., Extension of Extended Beta, Hypergeometric and Confluent Hypergeometric Functions, Honam Math. J., 36 (2) (2014) 357-385.
  • [14] Kulip M.A.H., Mohsen F.F., Barahmah S.S., Futher Extended Gamma and Beta Functions in Term of Generalized Wright Function, Electronic J. Uni. Aden Basic Appl. Sci., 1 (2) (2020) 78-83.
  • [15] Şahin R., Yağcı O., Yağbasan M.B., Kıymaz İ.O., Çetinkaya A., Further Generalizations of Gamma, Beta and Related Functions, J. Inequalities Spec. Func., 9 (4) (2018) 1-7.
  • [16] Mubeen S., Rahman G., Nisar K.S., Choi J., An Extended Beta Function and Its Properties, J. Math. Sci., 102 (7) (2017) 1545-1557.
  • [17] Abubakar U.M., A Study of Extended Beta and Associated Functions Connected to Fox-Wright Function, J. Frac. Calc. Appl., 12 (3) (13) (2021) 1-23.
  • [18] Yağcı, O., Şahin, R., Degenerate Pochhammer Symbol, Degenerate Sumudu Transform, and Degenerate Hypergeometric Function with Applications, Hacet. J. Math. Stat., 50 (5) (2021) 1448-1465.
  • [19] Şahin, R., Yağcı, O., A New Generalization of Pochhammer Symbol and its Applications, Appl. Math. Nonlinear Sci., 5 (1) (2020) 255-266.
  • [20] Şahin, R., & Yağcı, O., Fractional Calculus of the Extended Hypergeometric Function, Appl. Math. Nonlinear Sci., 5 (1) (2020) 369-384.
  • [21] Abdalla, M., Hidan, M., Boulaaras, S.M., Cherif, B.B., Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals, Math. Prob. Eng., 2021 (2011) 1-11.
  • [22] Saboor, A., Rahman, G., Ali, H., Nisar, K.S., Abdeljawad, T., Properties and Applications af a New Extended Gamma Function Involving Confluent Hypergeometric Function, J. Math., 2021 (2021) 1-12.
  • [23]Ata E., Modified Special Functions Defined by Generalized M-Series and their Properties, arXiv:2201.00867v1 [math.CA], (2022).
  • [24]Ata E., Generalized Beta Function Defined by Wright Function, arXiv:1803.03121v3 [math.CA], (2021).
  • [25]Debnath, L., Bhatta, D., Integral transforms and their applications, Third edition, CRC Pres, Boca Raton, London, New York, (2015) 143-398.
  • [26]Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, (1998) 103-109.
  • [27] Andrews G.E., Askey R., Roy R., Special functions, Cambridge university press, Cambridge, (1999) 1-60.
  • [28]Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential, Elsevier, North-Holland mathematics studies 204, Amsterdam, (2006) 1-68.

Generalized Gamma, Beta and Hypergeometric Functions Defined by Wright Function and Applications to Fractional Differential Equations

Year 2022, Volume: 43 Issue: 4, 684 - 695, 27.12.2022

Enes Ata , İ. Onur Kıymaz

https://doi.org/10.17776/csj.1005486
Cited By: 10

Abstract

When the literature is examined, it is seen that there are many studies on the generalizations of gamma, beta and hypergeometric functions. In this paper, new types of generalized gamma and beta functions are defined and examined using the Wright function. With the help of generalized beta function, new type of generalized Gauss and confluent hypergeometric functions are obtained. Furthermore, some properties of these functions such as integral representations, derivative formulas, Mellin transforms, Laplace transforms and transform formulas are determined. As examples, we obtained the solution of fractional differential equations involving the new generalized beta, Gauss hypergeometric and confluent hypergeometric functions. Finally, we presented their relationship with other generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions, which can be found in the literature.

Keywords

Beta function Wright function Gauss hypergeometric function Laplace transform fractional differential equation

References

  • [1] Chaudhry M.A., Zubair S.M., Generalized Incomplete Gamma Functions with Applications, J.Comput. Appl. Math., 55 (1) (1994) 99-124.
  • [2] Chaudhry M.A., Qadir A., Rafique M., Zubair S.M., Extension of Euler’s Beta Function, J. Comput. Appl. Math., 78 (1) (1997) 19-32.
  • [3] Chaudhry M.A., Qadir A., Srivastava H.M., Paris R.B., Extended Hypergeometric and Confluent Hypergeometric Functions, Appl. Math. Comput., 159 (2) (2004) 589-602.
  • [4] Ata E., Kıymaz İ.O., A Study on Certain Properties of Generalized Special Functions Defined by Fox-Wright Function, Appl. Math. Nonlinear Sci., 5 (1) (2020) 147-162.
  • [5] Özergin E., Özarslan M.A., Altın A., Extension of Gamma, Beta and Hypergeometric Functions, J. Comput. Appl. Math., 235 (16) (2011) 4601-4610.
  • [6] Lee D.M., Rathie A.K., Parmar R.K., Kim Y.S., Generalization of Extended Beta Function, Hypergeometric and Confluent Hypergeometric Functions, Honam Math. J., 33 (2) (2011) 187-206.
  • [7] Parmar R.K., A New Generalization of Gamma, Beta, Hypergeometric and Confluent Hypergeometric Functions, Le Matematiche, 68 (2) (2013) 33-52.
  • [8] Srivastava H.M., Agarwal P., Jain S., Generating Functions for the Generalized Gauss Hypergeometric Functions, Appl. Math. Comput., 247 (C) (2014) 348-352.
  • [9] Shadab M., Saime J., Choi J., An Extended Beta Function and Its Applications, J. Math. Sci., 103 (1) (2018) 235-251.
  • [10] Rahman G., Mubeen S., Nisar K.S., A New Generalization of Extended Beta and Hypergeometric Functions, J. Frac. Calc. Appl., 11 (2) (2020) 32-44.
  • [11] Çetinkaya A., Kıymaz İ.O., Agarwal P., Agarwal R.A., A Comparative Study on Generating Relations for Generalized Hypergeometric Functions via Generalized Fractional Operators, Adv. Differ. Equ., 2018 (1) (2018) 1-11.
  • [12] Goswami A., Jain S., Agarwal P., Aracı S., A Note on the New Extended Beta and Gauss Hypergeometric Functions, Appl. Math. Infor. Sci., 12 (1) (2018) 139-144.
  • [13] Choi J., Rathie A.K., Parmar R.K., Extension of Extended Beta, Hypergeometric and Confluent Hypergeometric Functions, Honam Math. J., 36 (2) (2014) 357-385.
  • [14] Kulip M.A.H., Mohsen F.F., Barahmah S.S., Futher Extended Gamma and Beta Functions in Term of Generalized Wright Function, Electronic J. Uni. Aden Basic Appl. Sci., 1 (2) (2020) 78-83.
  • [15] Şahin R., Yağcı O., Yağbasan M.B., Kıymaz İ.O., Çetinkaya A., Further Generalizations of Gamma, Beta and Related Functions, J. Inequalities Spec. Func., 9 (4) (2018) 1-7.
  • [16] Mubeen S., Rahman G., Nisar K.S., Choi J., An Extended Beta Function and Its Properties, J. Math. Sci., 102 (7) (2017) 1545-1557.
  • [17] Abubakar U.M., A Study of Extended Beta and Associated Functions Connected to Fox-Wright Function, J. Frac. Calc. Appl., 12 (3) (13) (2021) 1-23.
  • [18] Yağcı, O., Şahin, R., Degenerate Pochhammer Symbol, Degenerate Sumudu Transform, and Degenerate Hypergeometric Function with Applications, Hacet. J. Math. Stat., 50 (5) (2021) 1448-1465.
  • [19] Şahin, R., Yağcı, O., A New Generalization of Pochhammer Symbol and its Applications, Appl. Math. Nonlinear Sci., 5 (1) (2020) 255-266.
  • [20] Şahin, R., & Yağcı, O., Fractional Calculus of the Extended Hypergeometric Function, Appl. Math. Nonlinear Sci., 5 (1) (2020) 369-384.
  • [21] Abdalla, M., Hidan, M., Boulaaras, S.M., Cherif, B.B., Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals, Math. Prob. Eng., 2021 (2011) 1-11.
  • [22] Saboor, A., Rahman, G., Ali, H., Nisar, K.S., Abdeljawad, T., Properties and Applications af a New Extended Gamma Function Involving Confluent Hypergeometric Function, J. Math., 2021 (2021) 1-12.
  • [23]Ata E., Modified Special Functions Defined by Generalized M-Series and their Properties, arXiv:2201.00867v1 [math.CA], (2022).
  • [24]Ata E., Generalized Beta Function Defined by Wright Function, arXiv:1803.03121v3 [math.CA], (2021).
  • [25]Debnath, L., Bhatta, D., Integral transforms and their applications, Third edition, CRC Pres, Boca Raton, London, New York, (2015) 143-398.
  • [26]Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, (1998) 103-109.
  • [27] Andrews G.E., Askey R., Roy R., Special functions, Cambridge university press, Cambridge, (1999) 1-60.
  • [28]Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential, Elsevier, North-Holland mathematics studies 204, Amsterdam, (2006) 1-68.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Enes Ata 0000-0001-6893-8693

İ. Onur Kıymaz 0000-0003-2375-0202

Publication Date December 27, 2022
Submission Date October 6, 2021
Acceptance Date November 30, 2022
Published in Issue Year 2022Volume: 43 Issue: 4

Cite

APA Ata, E., & Kıymaz, İ. O. (2022). Generalized Gamma, Beta and Hypergeometric Functions Defined by Wright Function and Applications to Fractional Differential Equations. Cumhuriyet Science Journal, 43(4), 684-695. https://doi.org/10.17776/csj.1005486

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