Research Article

Year 2022,
Volume: 4 Issue: 2, 21 - 31, 31.12.2022
### Abstract

### Keywords

### Supporting Institution

### Project Number

### References

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.

Classical factorial function Classical Pochhammer symbol Gauss and confluent hypergeometric functions Two parameters factorial function and Two parameters Pochhammer symbol.

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- [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.

Year 2022,
Volume: 4 Issue: 2, 21 - 31, 31.12.2022
### Abstract

### Project Number

### References

NIL

- [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.

There are 21 citations in total.

Primary Language | English |
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Subjects | Mathematical Sciences |

Journal Section | Articles |

Authors | |

Project Number | NIL |

Early Pub Date | December 31, 2022 |

Publication Date | December 31, 2022 |

Acceptance Date | October 13, 2022 |

Published in Issue | Year 2022 Volume: 4 Issue: 2 |