New Generalized Hypergeometric Functions

New Generalized Hypergeometric Functions

The classical Gauss hypergeometric function and the Kumar confluent hypergeometric function are defined using a classical Pochammer symbol , and a factorial function. This research paper will present a two-parameter Pochhammer symbol, and discuss some of its properties such as recursive formulae and integral representation. In addition, the generalized Gauss and Kumar confluent hypergeometric functions are defined using a two-parameter Pochhammer symbol and two-parameter factorial function and some of the properties of the new generalized hypergeometric functions were also discussed.

___

  • [1] Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. National Bureau of Standards, Washington.
  • [2] Rehman, A., Mubeen, S., Ahmad, M. O. and Siddiqi, S. R. (2017). - Multiple Factorials with Applications. Punjab University Journal of Mathematics, (ISSN 1016-2526), Vol. 49(2) pp. 1-11.
  • [3] Mubeen, S. and Rehman, A. (2014). -Factorials. Journal of Inequalities and Special Functions, 5. No.3, pp. 14-20.
  • [4] Mubeen, S., Rehman, G. and Arshad, M. (2015). K-Gamma, K-Beta Matrix Functions and their Properties. J. Math. Comp. Sci., No. 5, pp. 647-657, ISSN:1927-5307.
  • [5] Thukral, A. K. (2014). Factorials of Real Negative and Imaginary Numbers – A New Perspective. Springer plus. 3:658 doi : 101186/2193-1801-3-658.
  • [6] Wolfram.com, “A Comprehensive Online Compendium of Formulas Involving the Special Functions of Mathematics”. http://functions.wolfram.com/constant/E/.
  • [7] Rafael, D. and Pariguan, E. (2005). On Hypergeometric Functions and K-Pochammer Symbol. arXiv:math/0405596.
  • [8] Gonzalez, I., Jiu, L. and Moll, V. H. (2015). Pochammer Symbol with Negative indices – A New Rule for the Method of Brackets. ArXiv: 1508.00056v1.*8.
  • [9] Milovanovic, G. V. and Petojevic, A. Generalised Factorials Functions, Numbers and Polynomials.
  • [10] Cattani, E. (2006). Three Lectures on Hypergeometric Functions. Department of Mathematics and Statistics, University of Massachusetts, Amherst, M.A 01003.
  • [11] Shrivastava, H. M., Cetinkaya, A. and Kiymaz, O. (2014). A Certain Generalized Pochammer Symbol and its Applications to Hypergeometric Functions. Journal of Applied Mathematics and Computation 226, 484 – 491.
  • [12] Sahin, R. and Yagci, O. (2020). A New Generalisation of Pochhammer Symbol and its Applications. Applied Mathematics and Nonlinear Sciences 5(1), pp. 255 – 266.
  • [13] Parmar, R. K and Raina, R. K. (2017). On the Extended Incomplete Pochhammer Symbols and Hypergeometric Functions”.
  • [14] Chaudary, M. A. and Zubair S. M. (1994). Generalized Incomplete Gamma Functions with Applications”. Journal of Computing and Applied Mathematics and. (55), pp. 99 – 124.
  • [15] Mubeen, S. and Rehman, A (2014). A Note on k-Gamma Function and Pochammer k-Symbol. Journal of Informatics and Mathematical Sciences. Vol. 6, No. 2, pp. 93 – 107. ISSN 0975 – 5748.
  • [16] Srivastava, R. (2013). Some Generalizations of Pochhammer’s Symbol and their Associated Families of Hypergeometric Functions and Hypergeometric Polynomials. Applied Mathematics and Information Sciences. 7, No. 6, 2195 – 2206.
  • [17] Petojevic, A. (2008). A Note about the Pochhammer Symbol. Mathematica Moravica. Vol. 12 – 1, pp. 37 – 42.
  • [18] Sahai, V. and Verma, A. (2016). On an Extension of the Generalized Pochhammer Symbol and its Applications to Hypergeometric Functions. Asian-European Journal of Mathematics. Vol. 9, No. 2, 1650064.
  • [19] Srivastava, H. M., Parmar, R. K. and Chopra, P. (2012). A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions. Axiom 1, pp. 2195 – 2206.
  • [20] Ozergin, E., Ozarslan, M. A. and Altin, A. (2011). Extension of Gamma, Beta and Hypergeometric Functions. Journal of Computational and Applied Mathematics. 235, pp. 4601 – 4610.
  • [21] Safdar, M., Rahman, G., Ulla, Z., Ghaffar, A. and Nissar, K. S. (2019). A New Extension of the Pochhammer Symbol and Application to Hypergeometric Functions. Journal of Applied and Computational Mathematics. 5 – 151. https://doi.ord/10.1007/s40819-019-0733-9.