Finite integral formula involving Aleph-function and generalized Mittag--Leffler function
Problemy analiza, Tome 9 (2020) no. 1, pp. 96-109.

Voir la notice de l'article provenant de la source Math-Net.Ru

The aim of this paper is to establish general definite integrals involving product of the Aleph function and the generalized Mittag-Leffler function with general arguments. This integral yields a number of known results as special cases. For the sake of illustration, several corollaries are also presented as special case of our main results.
Keywords: aleph-function, $i$-function, $h$-function, generalized Mittag–Leffler function.
@article{PA_2020_9_1_a7,
     author = {D. Kumar and F. Y. Ayant and A. Singh and P. K. Banerji},
     title = {Finite integral formula involving {Aleph-function} and generalized {Mittag--Leffler} function},
     journal = {Problemy analiza},
     pages = {96--109},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2020_9_1_a7/}
}
TY  - JOUR
AU  - D. Kumar
AU  - F. Y. Ayant
AU  - A. Singh
AU  - P. K. Banerji
TI  - Finite integral formula involving Aleph-function and generalized Mittag--Leffler function
JO  - Problemy analiza
PY  - 2020
SP  - 96
EP  - 109
VL  - 9
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2020_9_1_a7/
LA  - en
ID  - PA_2020_9_1_a7
ER  - 
%0 Journal Article
%A D. Kumar
%A F. Y. Ayant
%A A. Singh
%A P. K. Banerji
%T Finite integral formula involving Aleph-function and generalized Mittag--Leffler function
%J Problemy analiza
%D 2020
%P 96-109
%V 9
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2020_9_1_a7/
%G en
%F PA_2020_9_1_a7
D. Kumar; F. Y. Ayant; A. Singh; P. K. Banerji. Finite integral formula involving Aleph-function and generalized Mittag--Leffler function. Problemy analiza, Tome 9 (2020) no. 1, pp. 96-109. http://geodesic.mathdoc.fr/item/PA_2020_9_1_a7/

[1] Ayant F. Y., Kumar D., “Certain finite double integrals involving the hypergeometric function and Aleph-function”, International Journal of Mathematics Trends and Technology, 35:1 (2016), 49–55 | DOI | MR

[2] Brychkov Y. A., Handbook of Special Functions, Derivatives, Integrals, Series and other Formulas, CRC Press, Taylor Francis Group, Boca Raton–London–New York, 2008 | DOI | MR | Zbl

[3] Buschman R. G., Svrivastava H. M., “The $\overline{H}$- function associated with a certain class of Feynman integrals”, J. Phys A: Math. Gen., 23 (1990), 4707–4710 | DOI | Zbl

[4] Fox C., “The $G$ and $H$-functions as symmetrical Fourier Kernels”, Trans. Amer. Math. Soc., 98 (1961), 395–429 | MR | Zbl

[5] Gupta R. K., Shaktawat B. S., Kumar D., “Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function”, J. Raj. Acad. Phy. Sci., 15:3 (2016), 117–126 | MR | Zbl

[6] Kumar D., Choi J., Srivastava H. M., “Solution of a general family of fractional kinetic equations associated with the generalized Mittag–Leffler function”, Nonlinear Funct. Anal. Appl., 23:3 (2018), 455–471 | Zbl

[7] Kumar D., Saxena R. K., Ram J., “Finite integral formulas involving Aleph function”, Bol. Soc. Parana. Mat., 36:1 (2018), 177–193 | DOI | MR | Zbl

[8] Mathai A. M., Saxena R. K., Haubold H. J., The $H$-function: Theory and Applications, Springer, New York, 2010 | DOI | MR | Zbl

[9] Mittag-Leffler G. M., “Sur la nouvelle fonction $E_\alpha(x)$”, C.R. Acad. Sci. Paris, 137 (1903), 554–558

[10] Oberhettinger F., Tables of Mellin Transforms, Springer-Verlag, Berlin–Heidelberg–New York, 1974 | MR | Zbl

[11] Prabhakar T. R., “A singular integral equation with a generalize Mittag–Leffler function in the kernel”, Yokohama Math. J., 19 (1971), 7–15 | MR | Zbl

[12] Rainville E. D., Special Functions, The Macmillan Company, New York, 1960 | MR | Zbl

[13] Ram J., Kumar D., “Generalized fractional integration of the $\aleph$-function”, J. Raj. Acad. Phy. Sci., 10:4 (2011), 373–382 | MR

[14] Rathie A. K., “A new generalization of generalized hypergeometric functions”, Matematiche (Catania), 52:2 (1997), 297–310 | MR | Zbl

[15] Salim T. O., Faraj A. W., “A generalization of Mittag–Leffler function and integral operator associated with fractional calculus”, J. Fract. Calc. Appl., 3:5 (2012), 1–13

[16] Saxena V. P., “Formal solution of certain new pair of dual integral equations involving $H$-functions”, Proc. Nat. Acad. Sci. India Sect., A51 (1982), 366–375 | MR | Zbl

[17] Saxena R. K., Kumar D., “Generalized fractional calculus of the Aleph-function involving a general class of polynomials”, Acta Math. Sci. Ser. B (Engl. Ed.), 35:5 (2015), 1095–1110 | DOI | MR | Zbl

[18] Shukla A. K., Prajapati J. C., “On a generalization of Mittag–Leffler function and its properties”, J. Math. Anal. Appl., 336:2 (2007), 797–811 | DOI | MR | Zbl

[19] Srivastava H. M., Tomovski Z., “Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel”, Appl. Math. Comput., 211:1 (2009), 198–210 | DOI | MR | Zbl

[20] Südland N., Baumann B., Nonnenmacher T. F., Open problem: Who knows about the Aleph ($\aleph$)-functions?, Fract. Calc. Appl. Anal., 1:4 (1998), 401–402 | MR

[21] Südland N., Baumann B., Nonnenmacher T. F., “Fractional driftless Fokker-Planck equation with power law diffusion coefficients”, Computer Algebra in Scientific Computing, CASC 2001 (Konstanz, 2001), eds. V.G. Gangha, E.W. Mayr, W.G. Vorozhtsov, Springer, Berlin, 2001, 513–525 | MR

[22] Suthar D. L., Agarwal S., Kumar D., “Certain integrals involving the product of Gaussian hypergeometric function and Aleph function”, Honam Math. J., 41:1 (2019), 1–17 | DOI | MR | Zbl

[23] Wiman A., “Über den fundamental satz in der theorie der funktionen $E_\alpha(x)$”, Acta Math., 29 (1905), 191–201 | DOI | MR | Zbl