Boros Integral Involving the Product of Special Functions and the Incomplete I- Function
Abstract
In this present research, we developed a three parameter Boros integral formula for the incomplete I-function along with the generalized multi-index Mittag-
Leffler function (MLF) and Srivastava Polynomial. The derived outcomes are of a generic nature and may yield Boros integrals engaging the incomplete H(bar) - function and incomplete H- function as a specific instance.
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References
[1] Bhatter, S., Jangid, K., Meena, S., & Purohit, S.D. (2021). Certain Integral Formulae Involving
Incomplete I-Functions. Science & Technology Asia, 26(4). 84-95.
[2] Boros, G., & Moll, V.H. (1998). An integral with three parameters. Siam Review, 40(4), 972-980.
[3] Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (1954). Higher Transcendental Functions
3, McGraw–Hill, New York, Toronto & London . Reprinted: Krieger, Melbourne, Florida, 1981.
[4] Jangid, K., Bhatter, S., Meena, S. & Purohit, S.D. (2023). Certain classes of the incomplete Ifunctions
and their properties, Discontin. Nonlinearity Complex., 12(2), to appear.
[5] Kilbas, A.A., Marichev, O.I., & Samko, S.G. (1993). Fractional integrals and derivatives (theory
and applications).
[6] Kiryakova, V. S. (2000). Multiple (multiindex) Mittag–Leffler functions and relations to generalized
fractional calculus. J. Comp. Appl. Math., 118(1-2), 241-259.
[7] Kumar, D., Ayant, F., Nirwan, P., & Suthar, D. L. (2022). Boros integral involving the generalized
multi-index Mittag-Leffler function and incomplete I-functions. Research in Mathematics, 9(1), 1-7.
[8] Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag–Leffler function in
the kernel, Yokohama Math. J. 19 , 7–15.
[9] Qureshi, M. I., Quraishi, K. A., & Pal, R. (2011). Some definite integrals of Gradshteyn-Ryzhil and
other integrals. Glob. J. Sci. Front. Res., 11(4), 75-80.
[10] Rathie, A.K. (1997). A new generalization of generalized hypergeometric functions, LeMatematiche,
52(2), 297-310.
[11] Srivastava, H.M. & Singh, N.P. (1983). The integration of certain products of the multivariable
H-function with a general class of polynomials, Rend. Circ. Mat. Palermo, 32(2), 157-187.
[12] Saxena, R. K.,& Nishimoto, K. (2010).: N-fractional calculus of generalized Mittag-Leffler functions.
J. Fract. Calc, 37, 43-52.
[13] Saxena, R. K., & Nishimoto, K. (2010). : Further results on generalized Mittag-Leffler functions of
fractional calculus. J. Fract. Calc, 39, 29-41.
[14] Srivastava, H. M., & Tomovski, ˇZ. (2009). Fractional calculus with an integral operator containing
a generalized Mittag–Leffler function in the kernel. Appl. Math. Comput., 211(1), 198-210.
[15] Wiman, A. (1905). ¨Uber den fundamental Satz in der Theorie der Funktionen Eα(x), Acta Math.,
pp. 191–201.
Incomplete I-Functions. Science & Technology Asia, 26(4). 84-95.
[2] Boros, G., & Moll, V.H. (1998). An integral with three parameters. Siam Review, 40(4), 972-980.
[3] Erd´elyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (1954). Higher Transcendental Functions
3, McGraw–Hill, New York, Toronto & London . Reprinted: Krieger, Melbourne, Florida, 1981.
[4] Jangid, K., Bhatter, S., Meena, S. & Purohit, S.D. (2023). Certain classes of the incomplete Ifunctions
and their properties, Discontin. Nonlinearity Complex., 12(2), to appear.
[5] Kilbas, A.A., Marichev, O.I., & Samko, S.G. (1993). Fractional integrals and derivatives (theory
and applications).
[6] Kiryakova, V. S. (2000). Multiple (multiindex) Mittag–Leffler functions and relations to generalized
fractional calculus. J. Comp. Appl. Math., 118(1-2), 241-259.
[7] Kumar, D., Ayant, F., Nirwan, P., & Suthar, D. L. (2022). Boros integral involving the generalized
multi-index Mittag-Leffler function and incomplete I-functions. Research in Mathematics, 9(1), 1-7.
[8] Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag–Leffler function in
the kernel, Yokohama Math. J. 19 , 7–15.
[9] Qureshi, M. I., Quraishi, K. A., & Pal, R. (2011). Some definite integrals of Gradshteyn-Ryzhil and
other integrals. Glob. J. Sci. Front. Res., 11(4), 75-80.
[10] Rathie, A.K. (1997). A new generalization of generalized hypergeometric functions, LeMatematiche,
52(2), 297-310.
[11] Srivastava, H.M. & Singh, N.P. (1983). The integration of certain products of the multivariable
H-function with a general class of polynomials, Rend. Circ. Mat. Palermo, 32(2), 157-187.
[12] Saxena, R. K.,& Nishimoto, K. (2010).: N-fractional calculus of generalized Mittag-Leffler functions.
J. Fract. Calc, 37, 43-52.
[13] Saxena, R. K., & Nishimoto, K. (2010). : Further results on generalized Mittag-Leffler functions of
fractional calculus. J. Fract. Calc, 39, 29-41.
[14] Srivastava, H. M., & Tomovski, ˇZ. (2009). Fractional calculus with an integral operator containing
a generalized Mittag–Leffler function in the kernel. Appl. Math. Comput., 211(1), 198-210.
[15] Wiman, A. (1905). ¨Uber den fundamental Satz in der Theorie der Funktionen Eα(x), Acta Math.,
pp. 191–201.
Published
2025-02-08
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