Research Article
BibTex RIS Cite

Year 2021, Volume: 42 Issue: 3, 663 - 676, 24.09.2021

Umar Muhammad Abubakar , Muhammad Lawan Kaurangini

https://doi.org/10.17776/csj.840774
Cited By: 1

Abstract

References

  • [1] Chaudhry M.A., Zubair S.M., Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55 (1994), 199-124.
  • [2] Chaudhry M.A., Zubair S.M., On decomposition of generalized incomplete gamma functions with applications to Fourier transform, Journal of Computational and Applied Mathematics, 59 (1995) 253-284.
  • [3] Chaudhry M.A., Zubair S.M., On extension of generalized incomplete gamma functions with applications, Journal of Australian Mathematical Society Series B, 37 (1996) 392-404.
  • [4] Chaudhry M.A., Transformation of extended gamma function Γ_(0, 2)^(2, 0) [(B,X)] with applications to astrophysical thermonuclear functions, Astrophysics and Space Science, 262 (1999) 263-270.
  • [5] Kulip M.A.H., Mohsen F.F., Barahmah S.S., Further extended gamma and beta functions in term of generalized Wright functions, Electronic Journal of University of Aden for Basic and Applied Sciences, 1 (2) (2020) 78-83.
  • [6] Ata E., Kiymaz I.O., A study on certain properties of generalized special function defined by Fox – Wright function, Applied Mathematics and Nonlinear Sciences, 5 (1) (2020) 147-162.
  • [7] He F., Bakhet A., Abdullah M. and Hidan M., On the extended hypergeometric functions and their applications for the derivatives of the extended Jacobi matrix polynomials, Mathematical Problems in Engineering, (2020) 4268361..
  • [8] Sahin R., Yagci O., Fractional calculus of the extended hypergeometric function, Applied Mathematics and Nonlinear Sciences, 5 (1) (2020) 369-384.
  • [9] Nisar K.S., Suthar D.l., Agarwal A. and Purohit S.D., Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag-Leffler function, Advance in Difference Equations, 148 (2020) 1-14.
  • [10] Kim T., kim D.S., Note on the degenerate gamma function, Russian Journal of Mathematical Physics, 27 (3) (2020) 352-358.
  • [11] Tassaddiq A., An application of theory of distributions to the family of λ-generalized gamma function, Mathematics, 5 (6) (2020) 5839-5858.
  • [12] Khan N., Usman T., Aman M., Extended beta, hypergeometric and confluent hypergeometric function, Transactions of National Academy of Science of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, Issue Mathematics, 39 (1) (2019) 83-97.
  • [13] Parmar R.K., Pogany T.K., On the Mathieu-type series for the unified Gauss hypergeometric functions, Applicable Analysis and Discrete Mathematics, 14 (2020) 138-149.
  • [14] Tilahun K., Tadessee H., Suthar D.L., The extended Bessel-Maitland function and integral operators associated with fractional calculus, Journal of Mathematics, (2020) 7582063.
  • [15] Suthar D.L., Baleanu D., Purohit S.D. and Ucar E., Certain k-fractional operators and images forms of k-struve function, Mathematics, 5 (3) (2020) 1706-1719.
  • [16] Suthar D.L., Khan A.M., Alaria A., Puhohit S.D. and Singh J., Extended Bessel-Maitland function and its properties pertaining of integral transforms and fractional calculus, Mathematics, 5 (2) (2020) 1400-1414.
  • [17] Abubakar U.M., Kabara S.R., A note on a new extended gamma and beta functions and their properties, IOSR Journal of Mathematics, 15 (5) (2019) 1-6.
  • [18] Abubakar U.M., Kabara S.R., Some results on the extension of the extended beta function, IOSR Journal of Mathematics, 15 (5) (2019) 7-12.
  • [19] Abubakar U.M., New generalized beta function associated with the Fox-Wright function, Journal of Fractional Calculus and Application, 12 (2) (2021), 204-227
  • [20] Abubakar U.M., Kabara S.R., Lawan M.A. and Idris F.A., A new extension of modified gamma and beta functions, Cankaya University Journal of Science and Engineering, 18 (1), 9-23 (2021).
  • [21] Oraby A., Ahmed M., Khaled M., Ahmed E. and Magdy M., Generalization of beta functions in term of Mittag-Leffler function, Frontiers in Scientific Research and Tehnology, 1 (2020) 81-88.
  • [22] Wiman A., Uber den fundamenta satz under theorie de function E_(α_1 ) (x), Acta Mathematica, 29 (1950) 191-201.
  • [23] Wiman A., Uber die nullstellum de funktionen〖 E〗_(α_1 ) (x), Acta Mathematica, 29 (1950) 217-234.
  • [24]Mittag-Leffler G.M., Sur la nouvelle function〖 E〗_(α_1 ) (x), Comptes Rendus de I’Academie des Sciences Paris Series II, 11 (137) (1903) 537-539.
  • [25] Mittag-Leffler G.M., Soprala funzione 〖 E〗_(α_1 ) (x), Redicoti della Academia dei Lincei, V(13) (1904) 3-5.
  • [26] Mittag-Leffler G.M., Sur larepresentation analytique d’une function monogone (inquieme note), Acta Mathematica, 29 (1905) 237-252.
  • [27] Chaudhry M.A., Zubair S.M., On a class of incomplete gamma functions with application, Chapman & Hall / CRC, 2002.
  • [28] Sahin R., Yagci, O., A new generalization of pochammer symbol and its application, Applied Mathematics and Nonlinear Science, 5(1) (2020) 255-266.
  • [29] Mardina F., Mura A., Pagnini G., The M-Wright function in Time-fractional diffusion processes: A tutorial survey, International Journal of Differential Equations, (2010) 104505.
  • [30] Rahman G., Saboor A., Anjum Z., Nisar and Abdeljawad T.A., An extension of the Mittag-Leffler function and its associated properties, Advances in Mathematical Physics, (2020) 5792853.
  • [31] Chaudhry M.A., Rathie A.K., Parmar R.K. and Kim Y.S., Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation 159 (2004) 589-602.
  • [32] Mathai A.M., Haubold H.J., Special functions for applied Scientists, Springer, 2008.
  • [33] Rahman G., Mubeen S., Nisar K.S., A new generalization of extended beta and hypergeometric functions, Journal of Fractional Calculus and Applications, 11 (2) (2020) 32-44.
  • [34] Shadab M., Jabee S., Choi J., An extended beta function and its applications, Far East Journal of Mathematical Sciences, 103 (1) (2018) 235-251.
  • [35] Luo M-J., Milovanovic G.V., Agarwal P., Some results on the extended beta and extended hypergeometric functions, Applied Mathematics and Computation, 248 (2014) 631-651.
  • [36] Lee D.M., Rathie A.K., Parmar R.K. and Kim Y.S, Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Homam Mathematical Journal, 33(2) (2011) 187-206.
  • [37] Chaudhry M.A., Qadir A., Rafique M. and Zubair S.M., Extension of Euler’s beta function, Journal of Computational and Applied Mathematics, 78 (1997) 19-32.
  • [38] Abubakar U.M., A study of extended beta and associated functions connected to Fox-Wright function, 12th Symposium of the Fractional Calculus and Applications Group, 1st International (ONLINE) Conference in Mathematical Science and Fractional Calculus, 16-18 February 2021, 1-23.
  • [39] Abubakar, U.M., Patel, S., On a new generalized beta function defined by the generalized Wright function and its applications, Malaysian Journal of Computing, 6 (2) (2021) 851-870.
  • [40] Abubakar U.M., A new generalization of Gengenbauer polynomials, Journal of Informatics and Mathematical Sciences, 13(2) (2021) 119-128.

New extension of beta, Gauss and confluent hypergeometric functions

Year 2021, Volume: 42 Issue: 3, 663 - 676, 24.09.2021

Umar Muhammad Abubakar , Muhammad Lawan Kaurangini

https://doi.org/10.17776/csj.840774
Cited By: 1

Abstract

There are many extensions and generalizations of Gamma and Beta functions in the literature. However, a new extension of the extended Beta function B_(ζ〖, α〗_1)^(α_2;〖 m〗_1,〖 m〗_2 ) (a_1,a_2 ) was introduced and presented here because of its important properties. The new extended Beta function has symmetric property, integral representations, Mellin transform, inverse Mellin transform and statistical properties like Beta distribution, mean, variance, moment and cumulative distribution which ware also presented. Finally, the new extended Gauss and Confluent Hypergeometric functions with their propertied were introduced and presented.   

Keywords

Gamma function Beta function Beta distribution Pochhammer symbol Mellin transform

References

  • [1] Chaudhry M.A., Zubair S.M., Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55 (1994), 199-124.
  • [2] Chaudhry M.A., Zubair S.M., On decomposition of generalized incomplete gamma functions with applications to Fourier transform, Journal of Computational and Applied Mathematics, 59 (1995) 253-284.
  • [3] Chaudhry M.A., Zubair S.M., On extension of generalized incomplete gamma functions with applications, Journal of Australian Mathematical Society Series B, 37 (1996) 392-404.
  • [4] Chaudhry M.A., Transformation of extended gamma function Γ_(0, 2)^(2, 0) [(B,X)] with applications to astrophysical thermonuclear functions, Astrophysics and Space Science, 262 (1999) 263-270.
  • [5] Kulip M.A.H., Mohsen F.F., Barahmah S.S., Further extended gamma and beta functions in term of generalized Wright functions, Electronic Journal of University of Aden for Basic and Applied Sciences, 1 (2) (2020) 78-83.
  • [6] Ata E., Kiymaz I.O., A study on certain properties of generalized special function defined by Fox – Wright function, Applied Mathematics and Nonlinear Sciences, 5 (1) (2020) 147-162.
  • [7] He F., Bakhet A., Abdullah M. and Hidan M., On the extended hypergeometric functions and their applications for the derivatives of the extended Jacobi matrix polynomials, Mathematical Problems in Engineering, (2020) 4268361..
  • [8] Sahin R., Yagci O., Fractional calculus of the extended hypergeometric function, Applied Mathematics and Nonlinear Sciences, 5 (1) (2020) 369-384.
  • [9] Nisar K.S., Suthar D.l., Agarwal A. and Purohit S.D., Fractional calculus operators with Appell function kernels applied to Srivastava polynomials and extended Mittag-Leffler function, Advance in Difference Equations, 148 (2020) 1-14.
  • [10] Kim T., kim D.S., Note on the degenerate gamma function, Russian Journal of Mathematical Physics, 27 (3) (2020) 352-358.
  • [11] Tassaddiq A., An application of theory of distributions to the family of λ-generalized gamma function, Mathematics, 5 (6) (2020) 5839-5858.
  • [12] Khan N., Usman T., Aman M., Extended beta, hypergeometric and confluent hypergeometric function, Transactions of National Academy of Science of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, Issue Mathematics, 39 (1) (2019) 83-97.
  • [13] Parmar R.K., Pogany T.K., On the Mathieu-type series for the unified Gauss hypergeometric functions, Applicable Analysis and Discrete Mathematics, 14 (2020) 138-149.
  • [14] Tilahun K., Tadessee H., Suthar D.L., The extended Bessel-Maitland function and integral operators associated with fractional calculus, Journal of Mathematics, (2020) 7582063.
  • [15] Suthar D.L., Baleanu D., Purohit S.D. and Ucar E., Certain k-fractional operators and images forms of k-struve function, Mathematics, 5 (3) (2020) 1706-1719.
  • [16] Suthar D.L., Khan A.M., Alaria A., Puhohit S.D. and Singh J., Extended Bessel-Maitland function and its properties pertaining of integral transforms and fractional calculus, Mathematics, 5 (2) (2020) 1400-1414.
  • [17] Abubakar U.M., Kabara S.R., A note on a new extended gamma and beta functions and their properties, IOSR Journal of Mathematics, 15 (5) (2019) 1-6.
  • [18] Abubakar U.M., Kabara S.R., Some results on the extension of the extended beta function, IOSR Journal of Mathematics, 15 (5) (2019) 7-12.
  • [19] Abubakar U.M., New generalized beta function associated with the Fox-Wright function, Journal of Fractional Calculus and Application, 12 (2) (2021), 204-227
  • [20] Abubakar U.M., Kabara S.R., Lawan M.A. and Idris F.A., A new extension of modified gamma and beta functions, Cankaya University Journal of Science and Engineering, 18 (1), 9-23 (2021).
  • [21] Oraby A., Ahmed M., Khaled M., Ahmed E. and Magdy M., Generalization of beta functions in term of Mittag-Leffler function, Frontiers in Scientific Research and Tehnology, 1 (2020) 81-88.
  • [22] Wiman A., Uber den fundamenta satz under theorie de function E_(α_1 ) (x), Acta Mathematica, 29 (1950) 191-201.
  • [23] Wiman A., Uber die nullstellum de funktionen〖 E〗_(α_1 ) (x), Acta Mathematica, 29 (1950) 217-234.
  • [24]Mittag-Leffler G.M., Sur la nouvelle function〖 E〗_(α_1 ) (x), Comptes Rendus de I’Academie des Sciences Paris Series II, 11 (137) (1903) 537-539.
  • [25] Mittag-Leffler G.M., Soprala funzione 〖 E〗_(α_1 ) (x), Redicoti della Academia dei Lincei, V(13) (1904) 3-5.
  • [26] Mittag-Leffler G.M., Sur larepresentation analytique d’une function monogone (inquieme note), Acta Mathematica, 29 (1905) 237-252.
  • [27] Chaudhry M.A., Zubair S.M., On a class of incomplete gamma functions with application, Chapman & Hall / CRC, 2002.
  • [28] Sahin R., Yagci, O., A new generalization of pochammer symbol and its application, Applied Mathematics and Nonlinear Science, 5(1) (2020) 255-266.
  • [29] Mardina F., Mura A., Pagnini G., The M-Wright function in Time-fractional diffusion processes: A tutorial survey, International Journal of Differential Equations, (2010) 104505.
  • [30] Rahman G., Saboor A., Anjum Z., Nisar and Abdeljawad T.A., An extension of the Mittag-Leffler function and its associated properties, Advances in Mathematical Physics, (2020) 5792853.
  • [31] Chaudhry M.A., Rathie A.K., Parmar R.K. and Kim Y.S., Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Applied Mathematics and Computation 159 (2004) 589-602.
  • [32] Mathai A.M., Haubold H.J., Special functions for applied Scientists, Springer, 2008.
  • [33] Rahman G., Mubeen S., Nisar K.S., A new generalization of extended beta and hypergeometric functions, Journal of Fractional Calculus and Applications, 11 (2) (2020) 32-44.
  • [34] Shadab M., Jabee S., Choi J., An extended beta function and its applications, Far East Journal of Mathematical Sciences, 103 (1) (2018) 235-251.
  • [35] Luo M-J., Milovanovic G.V., Agarwal P., Some results on the extended beta and extended hypergeometric functions, Applied Mathematics and Computation, 248 (2014) 631-651.
  • [36] Lee D.M., Rathie A.K., Parmar R.K. and Kim Y.S, Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Homam Mathematical Journal, 33(2) (2011) 187-206.
  • [37] Chaudhry M.A., Qadir A., Rafique M. and Zubair S.M., Extension of Euler’s beta function, Journal of Computational and Applied Mathematics, 78 (1997) 19-32.
  • [38] Abubakar U.M., A study of extended beta and associated functions connected to Fox-Wright function, 12th Symposium of the Fractional Calculus and Applications Group, 1st International (ONLINE) Conference in Mathematical Science and Fractional Calculus, 16-18 February 2021, 1-23.
  • [39] Abubakar, U.M., Patel, S., On a new generalized beta function defined by the generalized Wright function and its applications, Malaysian Journal of Computing, 6 (2) (2021) 851-870.
  • [40] Abubakar U.M., A new generalization of Gengenbauer polynomials, Journal of Informatics and Mathematical Sciences, 13(2) (2021) 119-128.
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Umar Muhammad Abubakar 0000-0003-3935-4829

Muhammad Lawan Kaurangini 0000-0001-9144-9433

Publication Date September 24, 2021
Submission Date December 14, 2020
Acceptance Date May 27, 2021
Published in Issue Year 2021Volume: 42 Issue: 3

Cite

APA Abubakar, U. M., & Kaurangini, M. L. (2021). New extension of beta, Gauss and confluent hypergeometric functions. Cumhuriyet Science Journal, 42(3), 663-676. https://doi.org/10.17776/csj.840774

Cited By

Some Results on the v-Analogue of Gamma Function
Earthline Journal of Mathematical Sciences
https://doi.org/10.34198/ejms.10122.109123
 Download Cover Image