Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function

Umar Muhammad Abubakar

doi:10.54187/jnrs.947322

EN

Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function

Abstract

Special functions such as hypergeometric, zeta, Bessel, Whittaker, Struve, Airy, Weber-Hermite and Mittag-Leffler functions are obtained as a solution to complex differential equations in engineering, science and technology. In this work, generalized Euler-type integrals involving four parameters Mittag-Leffler function are proposed. Some special cases of this type of generalized integrals that are corresponding to well-known results in the literature are also inferred.

Keywords

Factorial function, Pochhemmer symbol, Beta function, Mittag-Leffler function, Euler-type integrals

Thanks

I WOULD LIKE TO THANK THE EDITOR AND THE REVIEWERS OF THIS JOURNAL FOR THIS OPPORTUNITY, THE AUTHOR ACKNOWLEDGE ANY FORM OF SUGGESTIONS

References

L.C. Andrews, Special functions of mathematics for engineers, SPIE Press, Bellingham, 1998.
E.D. Rainville, Special functions, Macmillan, New York, 1960.
Y.L. Luke, The special functions and their approximations, Academic Press, New York, 1969.
H.M. Srivastava, J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier Amsterdam, 2012.
I.N. Sneddon, Special functions of mathematical physics and chemistry, Edinburgh, London, New York,Interscience Publishers, New York, 1956.
K.B. Oldham, J.C. Myland, J. Spaanier, An atlas of functions: with equator, the atlas function calculator, Springer, New York, 2009.
U.M. Abubakar, S.R. Kabara, M.A. Lawan, F.A. Idris, A new extension of modified gamma and beta functions, Cankaya University Journal of Science and Engineering, 18 (1), (2021) 9-23.
U.M. Abubakar, New generalized beta function associated with the Fox-Wright function, Journal of Fractional Calculus and Application, 12 (2), (2021) 204-227.
R.K. Saxena, D.C. Gokhroo, Special functions, Jaipur Publishing House, Jaipur, 1987.
M.A. Chaudhry, S.M. Zubair, Generalized incomplete gamma functions with applications, Journal of Computational and Applied Mathematics, 55, (1994)199-124.

M.A. Chaudhry, S.M. Zubair, On decomposition of generalized incomplete gamma functions with applications to Fourier transform, Journal of Computational and Applied Mathematics, 59, (1995) 253-284.
M.A. Chaudhry, S.M. Zubair, On extension of generalized incomplete gamma functions with applications, Journal of Australian Mathematical Society Series B, 37, (1996) 392-404.
M.A. Chaudhry, 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.
U.M. Abubakar, S.R. Kabara, A note on a new extended gamma and beta functions and their properties, IOSR Journal of Mathematics, 15 (5), (2019) 1-6.
M.A. Chaudhry, A. Qadir, M. Rafique, S.M. Zubair, Extension of Euler’s beta function, Journal of Computational and Applied Mathematics, 78, (1997)19-32.
M.A. Chaudhry, S.M. Zubair, On a class of incomplete gamma functions with application, Chapman & Hall / CRC, New York, 2002.
U.M. Abubakar, S.R. Kabara, Some results on the extension of the extended beta function, IOSR Journal of Mathematics, 15 (5), (2019) 7-12.
M.A. Chaudhry, A. Qadir, H.M. Srivastava, R.B. Paris, Extended hyper geometric and confluent hyper geometric functions, Journal of Computational and Applied Mathematics, 159, (2004) 589-602.
D.M. Lee, A.K. Rathie, R.K. Parmar, Y.S. Kim, Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Homam Mathematical Journal, 33 (2), (2011)187-206.
U.M. Abubakar, A study of extended beta and associated functions connected to Fox-Wright function, Journal of Fractional Calculus and Applications, 12(3), (2021), No:13, 1-23.
M-J. Luo, G.V. Milovanovic, P. Agarwal, Some results on the extended beta and extended hypergeometric functions, Applied Mathematics and Computation, 248, (2014) 631-651.
G.M. Mittag-Leffler, Sur la nouvelle function〖 E〗_α (x), Comptes Rendus de I’Academie des Sciences Paris Series II, 11 (137), (1903) 537-539.
G.M. Mittag-Leffler, Soprala funzione E_α (x), Redicoti della Academia dei Lincei, V (13), (1904) 3-5.
G.M. Mittag-Leffler, Sur larepresentation analytique d’une function monogone (inquieme note), Acta Mathematica, 29, (1905) 237-252.
A. Wiman, Uber den fundamenta satz under theorie de function E_α (x), Acta Mathematica, 29, (1950) 191-201.
A. Wiman, Uber die nullstellum de funktionen〖 E〗_α (x), Acta Mathematica, 29, (1950) 217- 234.
T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal, 19, (1971) 7-15.
A.K. Shukla, J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties, Journal of Mathematical Analysis and Applications, 336, (2007) 797-811.
H.M. Srivastava, Z. Tomovski, Functional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Applied Mathematics and Computations, 211, (2009) 198-210.
S. Khan, B. Agarwal, M.A. Pathan, F. Muhammad, Evaluation of Euler type integral, Applied Mathematics and Computation, 15, (2007), 31-49.
S. Jabee, M. Shadab, R.B. Paris, Certain results on Euler-type integrals and their applications, Ramanujan Journal, 54, (2021), 245-260.
M. Ali, W.A. Khan, I.A. Khan, Study on double integral operator involving generalized Bessel-Maitland function, Palestine Journal of Mathematics, 9 (2), (2020), 991-998.
W.A. Khan, K.S. Nisar, M. Ahmad, Euler type integral operator involving K-Mittag-Leffler function, Boletim da Socidade Paranaense de Matematica, 38 (5), (2020), 165-174.
M. Ghaysauddin, W.A. Khan, L.N. Mishra, On Hankle type integral transform associated with Whittaker and hypergeometric functions, Thai Journal of Mathematics, 19 (2), (2021), 521-527.
A.P. Prudniko, Yu.A. Brychkov, O.I. Matichev, Integrals and series, Vol. 1, Gordan and Breach Science Publishers, New York, 1990.
W.A. Khan, M. Ahmad, Some Euler type beta function involving extended Mittag-Leffler function, Palestine Journal of Mathematics, 9 (2), (2020) 969-976.
S. Ahmed, M.A. Khan, Euler type integral involving generalized Mittag-Leffler function, Communication of Korean Mathematical Society, 29 (3), (2014) 479-487.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

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

Publication Date

December 31, 2021

Submission Date

June 3, 2021

Acceptance Date

December 25, 2021

Published in Issue

Year 2021 Volume: 10 Number: 3

DOI

https://doi.org/10.54187/jnrs.947322

IZ

https://izlik.org/JA68RG75WU

APA

Abubakar, U. M. (2021). Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. Journal of New Results in Science, 10(3), 1-10. https://doi.org/10.54187/jnrs.947322

AMA

1.Abubakar UM. Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. JNRS. 2021;10(3):1-10. doi:10.54187/jnrs.947322

Chicago

Abubakar, Umar Muhammad. 2021. “Some Results on Generalized Euler-Type Integrals Related to the Four Parameters Mittag-Leffler Function”. Journal of New Results in Science 10 (3): 1-10. https://doi.org/10.54187/jnrs.947322.

EndNote

Abubakar UM (December 1, 2021) Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. Journal of New Results in Science 10 3 1–10.

IEEE

[1]U. M. Abubakar, “Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function”, JNRS, vol. 10, no. 3, pp. 1–10, Dec. 2021, doi: 10.54187/jnrs.947322.

ISNAD

Abubakar, Umar Muhammad. “Some Results on Generalized Euler-Type Integrals Related to the Four Parameters Mittag-Leffler Function”. Journal of New Results in Science 10/3 (December 1, 2021): 1-10. https://doi.org/10.54187/jnrs.947322.

JAMA

1.Abubakar UM. Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. JNRS. 2021;10:1–10.

MLA

Abubakar, Umar Muhammad. “Some Results on Generalized Euler-Type Integrals Related to the Four Parameters Mittag-Leffler Function”. Journal of New Results in Science, vol. 10, no. 3, Dec. 2021, pp. 1-10, doi:10.54187/jnrs.947322.

Vancouver

1.Umar Muhammad Abubakar. Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. JNRS. 2021 Dec. 1;10(3):1-10. doi:10.54187/jnrs.947322