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New Generalized Hypergeometric Functions

Year 2022, Volume: 4 Issue: 2, 21 - 31, 31.12.2022
https://doi.org/10.54286/ikjm.1100753

Abstract

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.

Supporting Institution

NIL

Project Number

NIL

References

  • [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.
Year 2022, Volume: 4 Issue: 2, 21 - 31, 31.12.2022
https://doi.org/10.54286/ikjm.1100753

Abstract

Project Number

NIL

References

  • [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.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Salım Rabı'u Kabara

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

Cite

APA Kabara, S. R. (2022). New Generalized Hypergeometric Functions. Ikonion Journal of Mathematics, 4(2), 21-31. https://doi.org/10.54286/ikjm.1100753
AMA Kabara SR. New Generalized Hypergeometric Functions. ikjm. December 2022;4(2):21-31. doi:10.54286/ikjm.1100753
Chicago Kabara, Salım Rabı’u. “New Generalized Hypergeometric Functions”. Ikonion Journal of Mathematics 4, no. 2 (December 2022): 21-31. https://doi.org/10.54286/ikjm.1100753.
EndNote Kabara SR (December 1, 2022) New Generalized Hypergeometric Functions. Ikonion Journal of Mathematics 4 2 21–31.
IEEE S. R. Kabara, “New Generalized Hypergeometric Functions”, ikjm, vol. 4, no. 2, pp. 21–31, 2022, doi: 10.54286/ikjm.1100753.
ISNAD Kabara, Salım Rabı’u. “New Generalized Hypergeometric Functions”. Ikonion Journal of Mathematics 4/2 (December 2022), 21-31. https://doi.org/10.54286/ikjm.1100753.
JAMA Kabara SR. New Generalized Hypergeometric Functions. ikjm. 2022;4:21–31.
MLA Kabara, Salım Rabı’u. “New Generalized Hypergeometric Functions”. Ikonion Journal of Mathematics, vol. 4, no. 2, 2022, pp. 21-31, doi:10.54286/ikjm.1100753.
Vancouver Kabara SR. New Generalized Hypergeometric Functions. ikjm. 2022;4(2):21-3.