New extension of beta, Gauss and confluent hypergeometric functions

Umar Muhammad Abubakar; Muhammad Lawan Kaurangini

doi:10.17776/csj.840774

EN

New extension of beta, Gauss and confluent hypergeometric functions

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.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Umar Muhammad Abubakar ^*
0000-0003-3935-4829
Nigeria

Muhammad Lawan Kaurangini This is me
0000-0001-9144-9433
Nigeria

Publication Date

September 24, 2021

Submission Date

December 14, 2020

Acceptance Date

May 27, 2021

Published in Issue

Year 2021 Volume: 42 Number: 3

DOI

https://doi.org/10.17776/csj.840774

IZ

https://izlik.org/JA64ZX48SB

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

AMA

1.Abubakar UM, Kaurangini ML. New extension of beta, Gauss and confluent hypergeometric functions. CSJ. 2021;42(3):663-676. doi:10.17776/csj.840774

Chicago

Abubakar, Umar Muhammad, and Muhammad Lawan Kaurangini. 2021. “New Extension of Beta, Gauss and Confluent Hypergeometric Functions”. Cumhuriyet Science Journal 42 (3): 663-76. https://doi.org/10.17776/csj.840774.

EndNote

Abubakar UM, Kaurangini ML (September 1, 2021) New extension of beta, Gauss and confluent hypergeometric functions. Cumhuriyet Science Journal 42 3 663–676.

IEEE

[1]U. M. Abubakar and M. L. Kaurangini, “New extension of beta, Gauss and confluent hypergeometric functions”, CSJ, vol. 42, no. 3, pp. 663–676, Sept. 2021, doi: 10.17776/csj.840774.

ISNAD

Abubakar, Umar Muhammad - Kaurangini, Muhammad Lawan. “New Extension of Beta, Gauss and Confluent Hypergeometric Functions”. Cumhuriyet Science Journal 42/3 (September 1, 2021): 663-676. https://doi.org/10.17776/csj.840774.

JAMA

1.Abubakar UM, Kaurangini ML. New extension of beta, Gauss and confluent hypergeometric functions. CSJ. 2021;42:663–676.

MLA

Abubakar, Umar Muhammad, and Muhammad Lawan Kaurangini. “New Extension of Beta, Gauss and Confluent Hypergeometric Functions”. Cumhuriyet Science Journal, vol. 42, no. 3, Sept. 2021, pp. 663-76, doi:10.17776/csj.840774.

Vancouver

1.Umar Muhammad Abubakar, Muhammad Lawan Kaurangini. New extension of beta, Gauss and confluent hypergeometric functions. CSJ. 2021 Sep. 1;42(3):663-76. doi:10.17776/csj.840774

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