×

Certain unified integrals associated with product of the general class of polynomials and incomplete \(I\)-functions. (English) Zbl 1499.33054

Summary: We derive some novel integrals coupled with incomplete \(I\)-functions and the general class of polynomials and the results are demonstrated in various forms of incomplete \(I\)-functions. Further, we reconstruct new integrals in terms of Fox’s \(H\)-function, \(I\)-function and incomplete \(H\)-functions by putting the parameters values in the incomplete \(I\)-functions. The general class of polynomials can be expressed as orthogonal polynomials like Hermite polynomial and Jacobi polynomial by substituting some precise values to the parameters. Such changes in the general class of polynomials can evaluate some novel integrals that contain orthogonal polynomials and incomplete \(I\)-functions in terms of incomplete \(I\)-functions.

MSC:

33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bansal, MK; Choi, J., A note on pathway fractional integral formulas associated with the incomplete H-functions, Int. J. Appl. Comput. Math., 5, 133 (2019) · Zbl 1428.26010
[2] Bansal, MK; Jolly, N.; Jain, R.; Kumar, D., An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results, J. Anal., 27, 727-740 (2019) · Zbl 1426.33052
[3] Chaurasia, VBL; Kumar, D., The integration of certain product involving special functions, Sci. Ser. A Math. Sci., 19, 7-12 (2010) · Zbl 1244.26009
[4] Bansal, MK; Kumar, D.; Khan, I.; Singh, J.; Nisar, KS, Certain unified integrals associated with product of m-series and incomplete H-functions, Mathematics, 7, 1191 (2019)
[5] Srivastava, HM; Singh, NP, The integration of certain products of the multivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo, 32, 157-187 (1983) · Zbl 0497.33003
[6] Garg, M.; Mittal, S., On a new unified integral, Proc. Indian Acad. Sci. Math. Sci., 114, 99-101 (2004) · Zbl 1052.33011
[7] Dumitru, B.; Kumar, D.; Purohit, S., Generalized fractional integrals of product of two H-functions and a general class of polynomials, Int. J. Comput. Math., 93, 1320-1329 (2015) · Zbl 1345.26010
[8] Choi, J.; Agarwal, P., Certain unified integrals associated with Bessel functions, Bound. Value Probl., 2013, 95 (2013) · Zbl 1293.33003
[9] Choi, J.; Agarwal, P.; Mathur, S.; Purohit, SD, Certain new integral formulas involving the generalized Bessel functions, Bull. Korean Math. Soc., 4, 995-1003 (2014) · Zbl 1303.33002
[10] Epstein, B., Some applications of the mellin transform in statistics, Ann. Math. Stat., 19, 370-379 (1948) · Zbl 0032.29203
[11] Yao K.: Spherically invariant random processes: theory and applications. In Communications, Information and Network Security; The Springer International Series in Engineering and Computer Science (Communications and Information Theory); Bhargava V.K., Poor H.V., Tarokh V., Yoon S., Eds.; Springer: Berlin/Heidelberg, Germany, 2003; Volume 712, pp. 315-332 · Zbl 1377.60054
[12] Yao, K., Simon, M.K., Biglieri, E.: A unified theory on wireless communications fading statistics based on SIRP. In Proceedings of the Fifth IEEE Workshop on Signal Processing Advances in Wireless Communications, Lisbon, Portugal, 11-14 July 2004; pp. 135-139.
[13] Chaurasia, VBL; Kumar, D., Application of special functions and SIRP in wireless communication fading statistics, Glob. J. Sci. Front. Res., 10, 14-19 (2010)
[14] Prym, FE, Zur theorie der gamma function, J. fur die Reine und Angewandte Mathematik., 82, 165-172 (1877) · JFM 08.0168.01
[15] Tricomi, FG, Sulla funzione gamma incomplete, Annali di Matematica, 31, 1, 263-279 (1950) · Zbl 0040.18401
[16] Srivastava, HM; Choi, J., Zeta and q-Zeta Functions and Associated Series and Integrals (2012), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1239.33002
[17] Chaudhry, MA; Zubair, SM, On a Class of Incomplete Gamma Functions with Applications, 515 (2001), Hoboken, NJ: Taylor and Francis, Hoboken, NJ
[18] Bansal, MK; Kumar, D., On the integral operators pertaining to a family of incomplete I-functions, AIMS-Math., 5, 1247-1259 (2020) · Zbl 1485.33014
[19] Singh, Y.; Gill, V.; Singh, J.; Kumar, D.; Khan, I., Computable generalization of fractional kinetic equations with special functions, J. King Saud Univ. Sci., 33, 1, 101221 (2021)
[20] Srivastava, HM, A contour integral involving Fox’s H-Junction, Indian J. Math., 14, 1-6 (1972) · Zbl 0226.33016
[21] Singh, Y., A unified study of the inverse Laplace transform of Aleph-Function involving general class of polynomial and their associated properties, Int. J. Creat. Res. Thought., 6, 1, 400-406 (2018)
[22] Szegő, G.: Orthogonal polynomials, 4th edition. American Mathematical Society, Providence, Rohde Island. American Mathematical Society, Colloquium Publications, Vol. XXIII. 1975. · Zbl 0305.42011
[23] Oberhettinger, F., Tables of Mellin Transforms (1974), New York: Springer, New York · Zbl 0289.44003
[24] Chaurasia, VBL; Singh, Y., Marichev-Saigo-Maeda fraction integration operators of certain special functions, Gen. Math. Notes., 26, 1, 134-144 (2015)
[25] Khan, NU; Kashmin, T., Some integrals for the generalized Bessel-Maitland functions, Electron. J. Math. Anal. Appl., 4, 139-149 (2016) · Zbl 1463.33009
[26] Nisar, KS; Rahman, G.; Mubeen, S.; Arshad, M., Certain new integral formulas involving the generalized k-Bessel function, Commun. Numer. Anal., 2017, 84-90 (2017)
[27] Bansal, MK; Kumar, D.; Jain, R., Interrelationships between Marichev-Saigo-Maeda Fractional Integral Operators, the Laplace Transform and the H-Function, Int. J. Appl. Comput. Math., 5, 103 (2019) · Zbl 1418.33001
[28] Bansal, MK; Kumar, D.; Jain, R., A study of Marichev-Saigo-Maeda fractional integral operators associated with S-generalized Gauss hypergeometric function, KYUNGPOOK Math. J., 59, 433-443 (2019) · Zbl 1434.33012
[29] Bhatter, S.; Mathur, A.; Kumar, D.; Singh, J., New extension of fractional-calculus results associated with product of certain special functions, Int. J. Appl. Comput. Math., 7, 97 (2021) · Zbl 1499.26008
[30] Srivastava, HM; Bansal, MK; Harjule, P., A study of fractional integral operators involving a certain generalized multi-index Mittag-Leffler function, Math. Meth. Appl. Sci., 41, 6108-6121 (2018) · Zbl 1488.26023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.