Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications

Year 2021, Volume: 50 Issue: 5, 1448 - 1465, 15.10.2021

Oğuz Yağcı Recep Şahin

https://doi.org/10.15672/hujms.738030

Cited By: 3

Abstract

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.

Keywords

degenerate Pochhammer symbol, degenerate Sumudu transform, degenerate hypergeometric function, gamma function, degenerate gamma function, beta function, generalized hypergeometric function, Laplace transform, degenerate Laplace transform, Stieltjes transform, Laguerre transform, generating function, fractional calculus operator

References

• [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.