Research Article

Year 2021,
Volume: 18 Issue: 1, 9 - 23, 01.05.2021
### Abstract

### References

- [1] M.A. Chaudhry, S.M. Zubair, On a class of incomplete gamma functions with application, Chapman & Hall / CRC, 2002.
- [2] M.A. Chaudhry, A. Qadir, M. Rafique, S.M. Zubair, “Extension of Euler beta function,” Journal of Computational and Applied Mathematics, vol. 78, pp. 19-32, 1997.
- [3] D.M. Lee, A.K. Rathie, R.K. Parmar, Y.S Kim , “Generalized extended beta function, hypergeometric and confluent hypergeometrc functions,” Honam Mathematical Journal, Vol. 33 no. 2, pp. 187 – 206, 2011.
- [4] J. Choi, A.K. Pallie, R.K. Parmar, “Extension of extended beta, hypergeometric and confluent hypergeometric functions,” Honam Mathematical Journal., vol. 36, no. 2, pp. 357 – 385, 2014.
- [5] E. Ozergin, M.A. Ozarslan, A. Altin, “Extension of gamma, beta and hypergeometric function,” Journal of Computational and Applied Mathematics, vol. 235, pp. 4601 – 4610, 2011.
- [6] R.K. Parmar, “A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions,” Le Mathematiche, vol. LXVIII, no. II, pp. 33 – 54, 2013.
- [7] P. Agarwal, J. Choi, S. Jain, “Extended hypergeometric functions of two and three variable,” Communication of the Korean Mathematical Society, vol. 30, no. 4, pp. 403 – 414, 2015.
- [8] P.I. Pucheta, “An new extended beta function,” International Journal of Mathematics and Its Applications, vol. 5, no. 3-C, pp. 255 – 260, 2017.
- [9] M. Chand, H. Hachimi, R. Rani, “New extension of beta function and its applications,” International Journal of Mathematical Sciences, Article ID: 6451592, 25 pages, 2018.
- [10] K.S. Gehlot and K.S. Nisar, “Extension of two parameter gamma, beta functions and its properties,” Applications and Applied Mathematics: An International Journal, Special Issue 6, pp. 37 – 55, 2020.
- [11] E.C. Emmanuel, “On analytical review of the gamma functions,” Asian Research Journal of Current Science, vol. 2 no. 1, pp. 28 – 33, Article noARJOCS.163, 2020.
- [12] M. Ghayasuddin, M. Ali and R.B. Paris, “Certain new extension of beta and related functions,”, 2020. [13] M.A.H. Kulip, F.F. Mohsen and S.S. Barahmah, “Further extended gamma and beta functions in term of generalized Wright functions,” Electronic Journal of University of Aden for Basic and Applied Sciences, vol. 1, no. 2, pp. 78 – 83, 2020.
- [14] A. Ata, I.O. Kiyma, “A study on certain properties of generalized special function defined by Fox – Wright function,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 1, pp. 147 – 162, 2020.
- [15] F. He, A. Bakhet, M. Abdullah, M. Hidan, “On the extended hypergeometric functions and their applications for the derivatives of the extended Jacobi matrix polynomials,” Mathematical Problems in Engineering, Article ID: 4268361, 8 pages, 2020.
- [16] R. Sahin, O. Yagci, “Fractional calculus of the extended hypergeometric function,” Applied Mathematics and Nonlinear Sciences, vol. 5 no. 1, pp. 369 – 384, 2020.
- [17] K.S. Nisar, D.l. Suthar, A. Agarwal, S.D. Purohit, “Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag – Leffler function,” Advance in Difference Equations, vol. 148, 14 pages, 2020.
- [18] T. Kim, D.S. Kim, “Note on the degenerate gamma function,” Russian Journal of Mathematical Physics, vol. 27, no. 3, pp. 352 – 358, 2020.
- [19] A. Tassaddiq, “An application of theory of distributions to the family of λ – generalized gamma function,” Mathematics, vol. 5, no. 6, pp. 5839 – 5858, 2020.
- [20] N. Khan, T. Usman, M. Aman, “Extended beta, hypergeometric and confluent hypergeometric function,” Transactions of National Academy of Science of Azerbaijan. Series of Physical – Technical and Mathematical Sciences, Issue Mathematics, vol. 39, no. 1, pp. 83 – 97, 2019.
- [21] M. Gayasuddin, N. Khan, M. Ali, “A study of the extended beta, Gauss and confluent hypergeometric functions,” International Journal of Applied Mathematics, vol. 33, no. 10, pp. 1 – 13, 2020.
- [22] D. Baleanu, P. Agarwal, R.K. Parmar, M.M. Alqurashi, S. Salahshour, “Extension of the fractional derivative operator of the Riemann – Liouville,” Journal of Nonlinear Sciences and Applicatons, vol. 10, pp. 2914 – 2924, 2017.
- [23] P. Agarwal, F. Qi, m. Chand, G. Singh, “Some fractional differential equations involving generalized hypergeometric functions,” Journal of Applied Analysis, vol. 25, no. 1, pp. 37 – 44, 2019.
- [24] R.K. Parmar, T.K. Pogany, “On the Mathieu – type series for the unified Gauss hypergeometric functions,” Applicable Analysis and Discrete Mathematics, vol. 14, pp. 138 – 149, 2020.
- [25] A. Koraby, M. Ahmed, M. Khaled, E. Ahmed, M. Magdy, “Generalization of beta functions in term of Mittag – Leffler function,” Frontiers in Scientific Research and Tehnology, vol. 1, pp. 81 – 88, 2020.
- [26] K.Tilahun, H. Tadessee, D.L. Suthar, “The extended Bessel – Maitland function and integral operators associated with fractional calculus,” Journal of Mathematics, Article ID: 7582063, 8 pages, 2020.
- [27] D.L. Suthar, D. Baleanu, S.D. Purohit, E. Ucar, “Certain k – fractional operators and images forms of k – struve function,” Mathematics, vol. 5, no. 3, pp. 1706 – 1719, 2020.
- [28] D.L. Suthar, A.M. Khan, A. Alaria, S.D. Puhohit, J. Singh, “Extended Bessel – Maitland function and its properties pertaining of integral transforms and fractional calculus,” Mathematics, vol. 5, no. 2, pp. 1400 – 1414, 2020.
- [29] S. Joshi, E. Mittal and R.M. Pandey, “On Euler types interval Involving Mittag – Leffler functions,” Boletim da Sociedate Paranaense de Matematica, vol. 38, no. 2, pp. 125 – 134, 2020.
- [30] N.U. Khan and S.W. Khan, “A new extension of the Mittag – Leffler Function,” Plastine Journal Mathematics, vol. 9, no. 2, pp. 977 – 983, 2020.
- [31] M. Saif, A.H. Khan and K.S. Nisar, “Integral transform of extended Mittag – Leffler in term of Fox – Wright function,” Palestine Journal of Mathematics, vol. 9, no. 1, pp. 456 – 463, 2020. [32] K.S. Ghelot, “Differential equation of p – k Mittag – Leffler function,” Palestine Journal of Mathematics, vol. 9, no. 2, pp. 940 – 944, 2020.
- [33] P.I. Pucheta, “An extended p – k Mittag – Leffler function,” Palestine Journal of Mathematics, vol. 9, no. 2, pp. 785 – 791, 2020.
- [34] U.M. Abubakar and S.R. Kabara, “A note on a new extended gamma and beta functions and their properties,” IOSR Journal of Mathematics, vol. 15, no. 5, pp. 1 – 6, 2019.
- [35] U.M. Abubakar and S.R. Kabara, “Some results on the extension of the extended beta function,” IOSR Journal of Mathematics, vol. 15, no. 5, pp. 7 – 12, 2019.

Year 2021,
Volume: 18 Issue: 1, 9 - 23, 01.05.2021
### Abstract

### Keywords

### References

In this research paper, a new extension of modified Gamma and Beta functions is presented and various functional, symmetric, first and second summation relations, Mellin transforms and integral representations are obtained. Furthermore, mean, variance and moment generating function for the beta distribution of the new extension of the modified beta function are also obtained.

Gamma function beta function Mittag-Leffler function modified gamma function modified beta function beta distribution

- [1] M.A. Chaudhry, S.M. Zubair, On a class of incomplete gamma functions with application, Chapman & Hall / CRC, 2002.
- [2] M.A. Chaudhry, A. Qadir, M. Rafique, S.M. Zubair, “Extension of Euler beta function,” Journal of Computational and Applied Mathematics, vol. 78, pp. 19-32, 1997.
- [3] D.M. Lee, A.K. Rathie, R.K. Parmar, Y.S Kim , “Generalized extended beta function, hypergeometric and confluent hypergeometrc functions,” Honam Mathematical Journal, Vol. 33 no. 2, pp. 187 – 206, 2011.
- [4] J. Choi, A.K. Pallie, R.K. Parmar, “Extension of extended beta, hypergeometric and confluent hypergeometric functions,” Honam Mathematical Journal., vol. 36, no. 2, pp. 357 – 385, 2014.
- [5] E. Ozergin, M.A. Ozarslan, A. Altin, “Extension of gamma, beta and hypergeometric function,” Journal of Computational and Applied Mathematics, vol. 235, pp. 4601 – 4610, 2011.
- [6] R.K. Parmar, “A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions,” Le Mathematiche, vol. LXVIII, no. II, pp. 33 – 54, 2013.
- [7] P. Agarwal, J. Choi, S. Jain, “Extended hypergeometric functions of two and three variable,” Communication of the Korean Mathematical Society, vol. 30, no. 4, pp. 403 – 414, 2015.
- [8] P.I. Pucheta, “An new extended beta function,” International Journal of Mathematics and Its Applications, vol. 5, no. 3-C, pp. 255 – 260, 2017.
- [9] M. Chand, H. Hachimi, R. Rani, “New extension of beta function and its applications,” International Journal of Mathematical Sciences, Article ID: 6451592, 25 pages, 2018.
- [10] K.S. Gehlot and K.S. Nisar, “Extension of two parameter gamma, beta functions and its properties,” Applications and Applied Mathematics: An International Journal, Special Issue 6, pp. 37 – 55, 2020.
- [11] E.C. Emmanuel, “On analytical review of the gamma functions,” Asian Research Journal of Current Science, vol. 2 no. 1, pp. 28 – 33, Article noARJOCS.163, 2020.
- [12] M. Ghayasuddin, M. Ali and R.B. Paris, “Certain new extension of beta and related functions,”, 2020. [13] M.A.H. Kulip, F.F. Mohsen and S.S. Barahmah, “Further extended gamma and beta functions in term of generalized Wright functions,” Electronic Journal of University of Aden for Basic and Applied Sciences, vol. 1, no. 2, pp. 78 – 83, 2020.
- [14] A. Ata, I.O. Kiyma, “A study on certain properties of generalized special function defined by Fox – Wright function,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 1, pp. 147 – 162, 2020.
- [15] F. He, A. Bakhet, M. Abdullah, M. Hidan, “On the extended hypergeometric functions and their applications for the derivatives of the extended Jacobi matrix polynomials,” Mathematical Problems in Engineering, Article ID: 4268361, 8 pages, 2020.
- [16] R. Sahin, O. Yagci, “Fractional calculus of the extended hypergeometric function,” Applied Mathematics and Nonlinear Sciences, vol. 5 no. 1, pp. 369 – 384, 2020.
- [17] K.S. Nisar, D.l. Suthar, A. Agarwal, S.D. Purohit, “Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag – Leffler function,” Advance in Difference Equations, vol. 148, 14 pages, 2020.
- [18] T. Kim, D.S. Kim, “Note on the degenerate gamma function,” Russian Journal of Mathematical Physics, vol. 27, no. 3, pp. 352 – 358, 2020.
- [19] A. Tassaddiq, “An application of theory of distributions to the family of λ – generalized gamma function,” Mathematics, vol. 5, no. 6, pp. 5839 – 5858, 2020.
- [20] N. Khan, T. Usman, M. Aman, “Extended beta, hypergeometric and confluent hypergeometric function,” Transactions of National Academy of Science of Azerbaijan. Series of Physical – Technical and Mathematical Sciences, Issue Mathematics, vol. 39, no. 1, pp. 83 – 97, 2019.
- [21] M. Gayasuddin, N. Khan, M. Ali, “A study of the extended beta, Gauss and confluent hypergeometric functions,” International Journal of Applied Mathematics, vol. 33, no. 10, pp. 1 – 13, 2020.
- [22] D. Baleanu, P. Agarwal, R.K. Parmar, M.M. Alqurashi, S. Salahshour, “Extension of the fractional derivative operator of the Riemann – Liouville,” Journal of Nonlinear Sciences and Applicatons, vol. 10, pp. 2914 – 2924, 2017.
- [23] P. Agarwal, F. Qi, m. Chand, G. Singh, “Some fractional differential equations involving generalized hypergeometric functions,” Journal of Applied Analysis, vol. 25, no. 1, pp. 37 – 44, 2019.
- [24] R.K. Parmar, T.K. Pogany, “On the Mathieu – type series for the unified Gauss hypergeometric functions,” Applicable Analysis and Discrete Mathematics, vol. 14, pp. 138 – 149, 2020.
- [25] A. Koraby, M. Ahmed, M. Khaled, E. Ahmed, M. Magdy, “Generalization of beta functions in term of Mittag – Leffler function,” Frontiers in Scientific Research and Tehnology, vol. 1, pp. 81 – 88, 2020.
- [26] K.Tilahun, H. Tadessee, D.L. Suthar, “The extended Bessel – Maitland function and integral operators associated with fractional calculus,” Journal of Mathematics, Article ID: 7582063, 8 pages, 2020.
- [27] D.L. Suthar, D. Baleanu, S.D. Purohit, E. Ucar, “Certain k – fractional operators and images forms of k – struve function,” Mathematics, vol. 5, no. 3, pp. 1706 – 1719, 2020.
- [28] D.L. Suthar, A.M. Khan, A. Alaria, S.D. Puhohit, J. Singh, “Extended Bessel – Maitland function and its properties pertaining of integral transforms and fractional calculus,” Mathematics, vol. 5, no. 2, pp. 1400 – 1414, 2020.
- [29] S. Joshi, E. Mittal and R.M. Pandey, “On Euler types interval Involving Mittag – Leffler functions,” Boletim da Sociedate Paranaense de Matematica, vol. 38, no. 2, pp. 125 – 134, 2020.
- [30] N.U. Khan and S.W. Khan, “A new extension of the Mittag – Leffler Function,” Plastine Journal Mathematics, vol. 9, no. 2, pp. 977 – 983, 2020.
- [31] M. Saif, A.H. Khan and K.S. Nisar, “Integral transform of extended Mittag – Leffler in term of Fox – Wright function,” Palestine Journal of Mathematics, vol. 9, no. 1, pp. 456 – 463, 2020. [32] K.S. Ghelot, “Differential equation of p – k Mittag – Leffler function,” Palestine Journal of Mathematics, vol. 9, no. 2, pp. 940 – 944, 2020.
- [33] P.I. Pucheta, “An extended p – k Mittag – Leffler function,” Palestine Journal of Mathematics, vol. 9, no. 2, pp. 785 – 791, 2020.
- [34] U.M. Abubakar and S.R. Kabara, “A note on a new extended gamma and beta functions and their properties,” IOSR Journal of Mathematics, vol. 15, no. 5, pp. 1 – 6, 2019.
- [35] U.M. Abubakar and S.R. Kabara, “Some results on the extension of the extended beta function,” IOSR Journal of Mathematics, vol. 15, no. 5, pp. 7 – 12, 2019.

There are 33 citations in total.

Primary Language | English |
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Subjects | Engineering |

Journal Section | Articles |

Authors | |

Publication Date | May 1, 2021 |

Published in Issue | Year 2021 Volume: 18 Issue: 1 |