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

Year 2021, , 1 - 10, 31.12.2021

Umar Muhammad Abubakar

https://doi.org/10.54187/jnrs.947322
Cited By: 2

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.

Year 2021, , 1 - 10, 31.12.2021

Umar Muhammad Abubakar

https://doi.org/10.54187/jnrs.947322
Cited By: 2

Abstract

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.
There are 37 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Umar Muhammad Abubakar 0000-0003-3935-4829

Publication Date December 31, 2021
Published in Issue Year 2021

Cite

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 Abubakar UM. Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. JNRS. December 2021;10(3):1-10. doi:10.54187/jnrs.947322
Chicago 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, no. 3 (December 2021): 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 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, 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 2021), 1-10. https://doi.org/10.54187/jnrs.947322.
JAMA 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, 2021, pp. 1-10, doi:10.54187/jnrs.947322.
Vancouver Abubakar UM. Some results on generalized Euler-type integrals related to the four parameters Mittag-Leffler function. JNRS. 2021;10(3):1-10.

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Symmetry
https://doi.org/10.3390/sym15101880


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