On the integral operators pertaining to a family of incomplete \(I\)-functions. (English) Zbl 1485.33014
Summary: This paper introduces a new incomplete \(I\)-functions. The incomplete \(I\)-function is an extension of the \(I\)-function given by V. P. Saxena [Proc. Natl. Acad. Sci. India, Sect. A 52, 366–375 (1982; Zbl 0535.45001)] which is a extension of a familiar Fox’s \(H\)-function. Next, we find the several interesting classical integral transform of these functions and also find the some basic properties of incomplete \(I\)-function. Further, numerous special cases are obtained from our main results among which some are explicitly indicated. Incomplete special functions thus obtained consisting of probability theory has many potential applications which are also presented. Finally, we find the solution of non-homogeneous heat conduction equation in terms of Incomplete \(I\)-function.
MSC:
| 33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |
| 33B15 | Gamma, beta and polygamma functions |
| 44A10 | Laplace transform |
Keywords:
gamma function; incomplete gamma functions; incomplete \(I\)-functions; Mellin-Barnes type contour; integral transformsCitations:
Zbl 0535.45001References:
| [1] | V. P. Saxena, i>Formal solution of certain new pair of dual integral equations involving H-functions</i, Proc. Nat. Acad. Sci. India Sect. A, 52, 366-375 (1982) · Zbl 0535.45001 |
| [2] | M. Abramowitz, I. A. Stegun, <i>Handbook of Mathematical Functions with Formulas; Graphs; and Mathematical Tables</i>, Applied Mathematics Series, 55, National Bureau of Standards, Washington, D.C., 1964; Reprinted by Dover Publications,New York, 1965. · Zbl 0171.38503 |
| [3] | L. C. Andrews, <i>Special Functions for Engineers and Applied Mathematicians</i>, Macmillan Company, New York, 1985. |
| [4] | W. Magnus, F. Oberhettinger, R. P. Soni, <i>Formulas and Theorems for the Special Functions of Mathematical Physics</i>, Third Enlarged Edition, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber ucksichtingung der Anwendungsgebiete, Bd. 52, Springer-Verlag, Berlin, Heidelberg and New York, 1966. · Zbl 0143.08502 |
| [5] | H. M. Srivastava, B. R. K. Kashyap, <i>Special Functions in Queuing Theory and Related Stochastic Processes</i>, Academic Press, New York and London, 1982. · Zbl 0492.60089 |
| [6] | M. K. Bansal; D. Kumar; R. Jain, i>Interrelationships between Marichev-Saigo-Maeda fractional integral operators, the Laplace transform and the \(\bar{H} \)-Function</i, Int. J. Appl. Comput. Math, 5 (2019) · Zbl 1418.33001 |
| [7] | M. K. Bansal; N. Jolly; R. Jain, t al., <i>An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results</i, J. Anal., 27, 727-740 (2019) · Zbl 1426.33052 · doi:10.1007/s41478-018-0119-0 |
| [8] | M. K. Bansal; D. Kumar; R. Jain, i>A study of Marichev-Saigo-Maeda fractional integral operators associated with S-Generalized Gauss Hypergeometric Function</i, KYUNGPOOK Math. J., 59, 433-443 (2019) · Zbl 1434.33012 |
| [9] | H. C. Kang; C. P. An, i>Differentiation formulas of some hypergeometric functions with respect to all parameters</i, Appl. Math. Comput., 258, 454-464 (2015) · Zbl 1338.33018 |
| [10] | S. D. Lin; H. M. Srivastava; J. C. Yao, i>Some classes of generating relations associated with a family of the generalized Gauss type hypergeometric functions</i, Appl. Math. Inform. Sci., 9, 1731-1738 (2015) |
| [11] | S. D. Lin; H. M. Srivastava; M. M. Wong, i>Some applications of Srivastava’s theorem involving a certain family of generalized and extended hypergeometric polynomials</i, Filomat, 29, 1811-1819 (2015) · Zbl 1474.33037 · doi:10.2298/FIL1508811L |
| [12] | H. M. Srivastava; P. Agarwal; S. Jain, i>Generating functions for the generalized Gauss hypergeometric functions</i, Appl. Math. Comput., 247, 348-352 (2014) · Zbl 1338.33015 |
| [13] | H. M. Srivastava; A. C. etinkaya; I. O. Kıymaz, i>A certain generalized Pochhammer symbol and its applications to hypergeometric functions</i, Appl. Math. Comput., 226, 484-491 (2014) · Zbl 1354.33002 |
| [14] | R. Srivastava, i>Some classes of generating functions associated with a certain family of extended and generalized hypergeometric functions</i, Appl. Math. Comput., 243, 132-137 (2014) · Zbl 1335.33005 |
| [15] | R. Srivastava; N. E. Cho, i>Some extended Pochhammer symbols and their applications involving generalized hypergeometric polynomials</i, Appl. Math. Comput., 234, 277-285 (2014) · Zbl 1302.33007 |
| [16] | R. Srivastava, i>Some properties of a family of incomplete hypergeometric functions</i, Russian J. Math. Phys., 20, 121-128 (2013) · Zbl 1279.33006 · doi:10.1134/S1061920813010111 |
| [17] | R. Srivastava; N. E. Cho, i>Generating functions for a certain class of incomplete hypergeometric polynomials</i, Appl. Math. Comput., 219, 3219-3225 (2012) · Zbl 1309.33009 |
| [18] | H. M. Srivastava; M. A. Chaudhry; R. P. Agarwal, i>The incomplete Pochhammer symbols and their applications to hypergeometric and related functions</i, Integral Transforms Spec. Funct., 23, 659-683 (2012) · Zbl 1254.33004 · doi:10.1080/10652469.2011.623350 |
| [19] | H. M. Srivastava; M. K. Bansal; P. Harjule, i>A study of fractional integral operators involving a certain generalized multi-index Mittag-Leffler function</i, Math. Meth. Appl. Sci., 41, 6108-6121 (2018) · Zbl 1488.26023 · doi:10.1002/mma.5122 |
| [20] | M. K. Bansal; D. Kumar; I. Khan, t al., <i>Certain unified integrals associated with product of M-series and incomplete H-functions</i, Mathematics, 7 (2019) |
| [21] | J. Singh, D. Kumar, M. K. Bansal, <i>Solution of nonlinear differential equation and special functions</i>, Math. Meth. Appl. Sci.,(2019) |
| [22] | I. S. Gradshteyn, I. M. Ryzhik, <i>Table of integrals, series, and |
| [23] | L. Debnath, D. Bhatta, <i>Integral Transforms and Their Applications</i>, Third edition, Chapman and Hall (CRC Press), Taylor and Francis Group, London and New York, 2014. · Zbl 1113.44001 |
| [24] | A. Erdélyi, W. Magnus, F. Oberhettinger, et al., <i>Higher Transcendental Functions</i> (Vols. I and II), McGraw-Hill Book Company, New York, Toronto and London, 1954. · Zbl 0051.30303 |
| [25] | I. N. Sneddon, <i>The Use of Integral Transforms</i>, Tata McGrawHill, New Delhi, India, 1979. |
| [26] | H. M. Srivastava; R. K. Saxena; R. K. Parmar, i>Some families of the incomplete H-Functions and the incomplete \(\bar{H} \)-Functions and associated integral transforms and operators of fractional calculus with applications</i, Russ. J. Math. Phys., 25, 116-138 (2018) · Zbl 1392.33008 · doi:10.1134/S1061920818010119 |
| [27] | M. K. Bansal; J. Choi, i>A note on pathway fractional integral formulas associated with the incomplete H</i>-<i>Functions</i, Int. J. Appl. Comput. Math, 5 (2019) · Zbl 1428.26010 |
| [28] | H. M. Srivastava, K. C. Gupta, S. P. Goyal, <i>The H-Functions of one and two variables with applications</i>, South Asian Publishers, New Delhi and Madras, 1982. · Zbl 0506.33007 |
| [29] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, <i>Theory and applications of fractional differential equations</i>, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. · Zbl 1092.45003 |
| [30] | M. A. Chaudhry; A. Qadir, i>Incomplete exponential and hypergeometric functions with applications to non-central χ<sup>2</sup><i>-Distribution</i, Comm. Statist. Theory Methods, 34, 525-535 (2002) · Zbl 1124.33003 |
| [31] | R. V. Churchill, <i>Fourier Series and Boundary Values Problems</i>, McGraw-Hill Book Co. New York, 1942. |
| [32] | G. Szegö, <i>Orthogonal Polynomials</i>, Fourth edition, Amererican Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, Rhode Island, 1975. · Zbl 0305.42011 |
| [33] | H. M. Srivastava, H. L. Manocha, <i>A Treatise on Generating Functions</i>, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984. · Zbl 0535.33001 |
| [34] | V. B. L. Chaurasia, i>The H-function and temperature in a non homogenous bar</i, Proc. Indian Acad Sci., 85, 99-103 (1977) · Zbl 0358.33005 · doi:10.1007/BF03046816 |
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.
