[1] M. Abramowitz and I.A. Stegun (Eds), Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied
Mathematics Series 55, 10th printing, Dover Publications, New York andWashington,
1972.
[2] J. Choi and P. Agarwal, Certain class of generating functions for the incomplete
hypergeometric functions, Abstr. Appl. Anal. 2014, 5 pages, 2014.
[3] J. Choi and P. Agarwal, Certain integral transforms for the incomplete functions,
Appl. Math. Inf. Sci. 9 (4), 2161–2167, 2015.
[4] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Third edition,
Chapman and Hall (CRC Press), Taylor and Francis Group, London and New York,
2014.
[5] A.K. Golmankhaneh and C. Tunç, Sumudu transform in fractal calculus, Appl. Math.
Comput. 350, 386–401, 2019.
[6] R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., River Edge, NJ, 2000.
[7] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Frac-
tional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Sci-
ence B.V., Amsterdam, 2006.
[8] T. Kim and D.S. Kim, Degenerate Laplace transform and degenerate gamma function,
Russ. J. Math. Phys. 24 (2), 241–248, 2017.
[9] A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-function, Theory and Applica-
tions, Springer, New York, 2010.
[10] F. Qi and B.-N. Guo, Relations among Bell polynomials, central factorial numbers,
and central Bell polynomials, Math. Sci. Appl. E-Notes 7 (2), 191–194, 2019.
[11] F. Qi, D.-W. Niu, D. Lim and B.-N. Guo, Closed formulas and identities for the Bell
polynomials and falling factorials, Contrib. Discrete Math. 14 (2), 1–11, 2019.
[12] F. Qi, D.-W. Niu, D. Lim and Y.-H. Yao, Special values of the Bell polynomials of the
second kind for some sequences and functions, J. Math. Anal. Appl. 491 (2), 124382,
2020.
[13] F. Qi, X.-T. Shi and F.-F. Liu, Several identities involving the falling and rising
factorials and the Cauchy, Lah, and Stirling numbers, Acta Univ. Sapientiae Math.
8 (2), 282–297, 2016.
[14] F. Qi, G.-S. Wu and B.-N. Guo, An alternative proof of a closed formula for central
factorial numbers of the second kind, Turkish J. Anal. Number Theory, 7 (2), 56–58,
2019.
[15] E.D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
[16] M. Safdar, G. Rahman, Z. Ullah, A. Ghaffar and K.S. Nisar, A New Extension of the
Pochhammer Symbol and Its Application to Hypergeometric Functions, Int. J. Appl.
Comput. Math. 5 (6), 151, 2019.
[17] H.M. Srivastava, P. Agarwal and S. Jain, Generating functions for the generalized
Gauss hypergeometric functions, Appl. Math. Comput. 247, 348–352, 2014.
[18] H.M. Srivastava, A. Çetinkaya and İ.O. Kıymaz, A certain generalized Pochhammer
symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226,
484–491, 2014.
[19] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted
Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chich-
ester, Brisbane and Toronto 1984.
[20] H.M. Srivastava, G. Rahman and K.S. Nisar, Some Extensions of the Pochhammer
Symbol and the Associated Hypergeometric Functions, Iran. J. Sci. Technol. Trans. A
Sci. 43 (5), 2601–2606, 2019.
[21] H.M. Srivastava and R.K. Saxena, Operators of fractional integration and their ap-
plications, Appl. Math. Comput. 118 (1), 1–52, 2001.
[22] R. Şahin and O. Yağcı, A New Generalization of Pochhammer Symbol and Its Appli-
cations, Appl. Math. Nonlinear Sci. 5 (1), 255–266, 2020.
[23] R. Şahin and O. Yağcı, Fractional Calculus of the Extended Hypergeometric Function,
Appl. Math. Nonlinear Sci. 5 (1), 369–384, 2020.
[24] G.K. Watugala, Sumudu transform: a new integral transform to solve differential
equations and control engineering problems, Internat. J. Math. Ed. Sci. Tech. 24 (1),
35–43, 1993.
[25] G.K. Watugala, Sumudu transform—a new integral transform to solve differential
equations and control engineering problems, Math. Eng. Ind. 6 (4), 319–329, 1998.
[26] G.K. Watugala, The Sumudu transform for functions of two variables, Math. Eng.
Ind. 8 (4), 293–302, 2002.
Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications
Year 2021,
Volume: 50 Issue: 5, 1448 - 1465, 15.10.2021
In the paper, we first define a degenerate Pochhammer symbol by using the degenerate gamma function and investigate its properties. By using the degenerate Pochhammer symbol, we introduce and investigate a degenerate hypergeometric function. We also define a degenerate Sumudu transform and investigate its properties by using degenerate exponential function. Finally, we give certain the integral representations, derivative formulas, integral transforms, factional calculus applications, and generating functions of the degenerate hypergeometric function.
[1] M. Abramowitz and I.A. Stegun (Eds), Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied
Mathematics Series 55, 10th printing, Dover Publications, New York andWashington,
1972.
[2] J. Choi and P. Agarwal, Certain class of generating functions for the incomplete
hypergeometric functions, Abstr. Appl. Anal. 2014, 5 pages, 2014.
[3] J. Choi and P. Agarwal, Certain integral transforms for the incomplete functions,
Appl. Math. Inf. Sci. 9 (4), 2161–2167, 2015.
[4] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Third edition,
Chapman and Hall (CRC Press), Taylor and Francis Group, London and New York,
2014.
[5] A.K. Golmankhaneh and C. Tunç, Sumudu transform in fractal calculus, Appl. Math.
Comput. 350, 386–401, 2019.
[6] R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., River Edge, NJ, 2000.
[7] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Frac-
tional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Sci-
ence B.V., Amsterdam, 2006.
[8] T. Kim and D.S. Kim, Degenerate Laplace transform and degenerate gamma function,
Russ. J. Math. Phys. 24 (2), 241–248, 2017.
[9] A.M. Mathai, R.K. Saxena and H.J. Haubold, The H-function, Theory and Applica-
tions, Springer, New York, 2010.
[10] F. Qi and B.-N. Guo, Relations among Bell polynomials, central factorial numbers,
and central Bell polynomials, Math. Sci. Appl. E-Notes 7 (2), 191–194, 2019.
[11] F. Qi, D.-W. Niu, D. Lim and B.-N. Guo, Closed formulas and identities for the Bell
polynomials and falling factorials, Contrib. Discrete Math. 14 (2), 1–11, 2019.
[12] F. Qi, D.-W. Niu, D. Lim and Y.-H. Yao, Special values of the Bell polynomials of the
second kind for some sequences and functions, J. Math. Anal. Appl. 491 (2), 124382,
2020.
[13] F. Qi, X.-T. Shi and F.-F. Liu, Several identities involving the falling and rising
factorials and the Cauchy, Lah, and Stirling numbers, Acta Univ. Sapientiae Math.
8 (2), 282–297, 2016.
[14] F. Qi, G.-S. Wu and B.-N. Guo, An alternative proof of a closed formula for central
factorial numbers of the second kind, Turkish J. Anal. Number Theory, 7 (2), 56–58,
2019.
[15] E.D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
[16] M. Safdar, G. Rahman, Z. Ullah, A. Ghaffar and K.S. Nisar, A New Extension of the
Pochhammer Symbol and Its Application to Hypergeometric Functions, Int. J. Appl.
Comput. Math. 5 (6), 151, 2019.
[17] H.M. Srivastava, P. Agarwal and S. Jain, Generating functions for the generalized
Gauss hypergeometric functions, Appl. Math. Comput. 247, 348–352, 2014.
[18] H.M. Srivastava, A. Çetinkaya and İ.O. Kıymaz, A certain generalized Pochhammer
symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226,
484–491, 2014.
[19] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted
Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chich-
ester, Brisbane and Toronto 1984.
[20] H.M. Srivastava, G. Rahman and K.S. Nisar, Some Extensions of the Pochhammer
Symbol and the Associated Hypergeometric Functions, Iran. J. Sci. Technol. Trans. A
Sci. 43 (5), 2601–2606, 2019.
[21] H.M. Srivastava and R.K. Saxena, Operators of fractional integration and their ap-
plications, Appl. Math. Comput. 118 (1), 1–52, 2001.
[22] R. Şahin and O. Yağcı, A New Generalization of Pochhammer Symbol and Its Appli-
cations, Appl. Math. Nonlinear Sci. 5 (1), 255–266, 2020.
[23] R. Şahin and O. Yağcı, Fractional Calculus of the Extended Hypergeometric Function,
Appl. Math. Nonlinear Sci. 5 (1), 369–384, 2020.
[24] G.K. Watugala, Sumudu transform: a new integral transform to solve differential
equations and control engineering problems, Internat. J. Math. Ed. Sci. Tech. 24 (1),
35–43, 1993.
[25] G.K. Watugala, Sumudu transform—a new integral transform to solve differential
equations and control engineering problems, Math. Eng. Ind. 6 (4), 319–329, 1998.
[26] G.K. Watugala, The Sumudu transform for functions of two variables, Math. Eng.
Ind. 8 (4), 293–302, 2002.
Yağcı, O., & Şahin, R. (2021). Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications. Hacettepe Journal of Mathematics and Statistics, 50(5), 1448-1465. https://doi.org/10.15672/hujms.738030
AMA
Yağcı O, Şahin R. Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications. Hacettepe Journal of Mathematics and Statistics. October 2021;50(5):1448-1465. doi:10.15672/hujms.738030
Chicago
Yağcı, Oğuz, and Recep Şahin. “Degenerate Pochhammer Symbol, Degenerate Sumudu Transform, and Degenerate Hypergeometric Function With Applications”. Hacettepe Journal of Mathematics and Statistics 50, no. 5 (October 2021): 1448-65. https://doi.org/10.15672/hujms.738030.
EndNote
Yağcı O, Şahin R (October 1, 2021) Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications. Hacettepe Journal of Mathematics and Statistics 50 5 1448–1465.
IEEE
O. Yağcı and R. Şahin, “Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, pp. 1448–1465, 2021, doi: 10.15672/hujms.738030.
ISNAD
Yağcı, Oğuz - Şahin, Recep. “Degenerate Pochhammer Symbol, Degenerate Sumudu Transform, and Degenerate Hypergeometric Function With Applications”. Hacettepe Journal of Mathematics and Statistics 50/5 (October 2021), 1448-1465. https://doi.org/10.15672/hujms.738030.
JAMA
Yağcı O, Şahin R. Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications. Hacettepe Journal of Mathematics and Statistics. 2021;50:1448–1465.
MLA
Yağcı, Oğuz and Recep Şahin. “Degenerate Pochhammer Symbol, Degenerate Sumudu Transform, and Degenerate Hypergeometric Function With Applications”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, 2021, pp. 1448-65, doi:10.15672/hujms.738030.
Vancouver
Yağcı O, Şahin R. Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications. Hacettepe Journal of Mathematics and Statistics. 2021;50(5):1448-65.