{"id":1118,"date":"2018-12-03T11:14:32","date_gmt":"2018-12-03T11:14:32","guid":{"rendered":"http:\/\/www.gyanvihar.org\/journals\/?p=1118"},"modified":"2019-05-27T08:58:45","modified_gmt":"2019-05-27T08:58:45","slug":"fractional-derivative-pertaining-to-i-function-and-lauricella-function","status":"publish","type":"post","link":"https:\/\/www.gyanvihar.org\/journals\/fractional-derivative-pertaining-to-i-function-and-lauricella-function\/","title":{"rendered":"FRACTIONAL DERIVATIVE PERTAINING TO I-FUNCTION AND LAURICELLA FUNCTION"},"content":{"rendered":"<p style=\"text-align: center\"><span style=\"font-family: 'times new roman', times, serif\"><strong>*HARSHITA GARG AND **ASHOK SINGH SHEKHAWAT <\/strong><\/span><\/p>\n<p style=\"text-align: center\"><span style=\"font-family: 'times new roman', times, serif\">*Suresh Gyan Vihar University, Jagatpura, Jaipur, Rajasthan, India<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"font-family: 'times new roman', times, serif\">**Arya College Of Engineering And Information Technology, Jaipur, Rajasthan, India<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>\u00a0<\/strong><strong>Abstract <\/strong>:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">The object of this paper is to obtain a fractional derivative of I- function associated with generalize Lauricella\u00a0 functions and general \u00a0\u00a0class of multivariable polynomials.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong><em>\u00a0Key words<\/em><\/strong><em>:<\/em> Fractional derivative operator, I-function, Lauricella function, general class of \u00a0\u00a0multivariable \u00a0\u00a0\u00a0polynomials.\u00a0<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0<\/span><span style=\"font-family: 'times new roman', times, serif\"><strong>INTRODUCTION: <\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">The I- function given by Saxena [4] is \u00a0\u00a0\u00a0\u00a0represented and defined as following:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong><u>\u00a0<\/u><\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1151\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/39.png\" alt=\"\" width=\"663\" height=\"428\" srcset=\"https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/39.png 663w, https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/39-624x403.png 624w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0\u00a0<\/span><span style=\"font-family: 'times new roman', times, serif\">p<sub>i<\/sub> (i= 1,\u2026.,R), q<sub>i<\/sub> (i= 1,\u2026.,R), e, f are integers satisfying 0 \u00a3 f \u00a3 p<sub>i<\/sub>, 0 \u00a3 e \u00a3 q<sub>i<\/sub> (i= 1,\u2026.,R); R is finite, \u03b1<sub>j, <\/sub>\u03b2<sub>j,\u00a0 <\/sub>\u03b1<sub>ji, <\/sub>\u03b2<sub>ji,\u00a0 <\/sub>are real and positive; a<sub>j, <\/sub>b<sub>j,\u00a0 <\/sub>a<sub>ji, <\/sub>b<sub>ji <\/sub>are complex numbers and \u00a3 is the path of integration separating the increasing and decreasing sequences of poles of the integrand.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">The integral converges if<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1153\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/40.png\" alt=\"\" width=\"663\" height=\"275\" srcset=\"https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/40.png 663w, https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/40-624x259.png 624w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/>\u00a0\u00a0\u00a0\u00a0<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 ,<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0<\/span><\/p>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1159\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/41.png\" alt=\"\" width=\"663\" height=\"924\" srcset=\"https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/41.png 663w, https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/41-624x870.png 624w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0 Now following shorthand notations given by Srivastava and Daoust [6] denote the generalized Lauricella function of several complex variables.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 =\u00a0\u00a0 \u00a0F \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026 (1.9)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/p>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1161\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/42.png\" alt=\"\" width=\"663\" height=\"836\" srcset=\"https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/42.png 663w, https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/42-624x787.png 624w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>Proof<\/strong><strong>: <\/strong>In order to prove (2.1), express the I-function in terms of Mellin-Barnes type of contour integrals by (1.1) and Lauricella function by (1.7) and general class of polynomials given by (1.4), then collecting the powers of (x-u) and (v-x). Finally making use of the result (1.6), we get the main result (2.1).<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>PARTICULAR CASES:<\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">If we take R=1 in (1.1), then I-function breaks into well known Fox\u2019s H-function and consequently there hold the following result:<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>\u00a0<\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1167\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/43.png\" alt=\"\" width=\"663\" height=\"376\" srcset=\"https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/43.png 663w, https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/43-624x354.png 624w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><span style=\"font-family: 'times new roman', times, serif\">Valid under the conditions surrounding (2.1)<\/span><\/p>\n<ul style=\"text-align: justify\">\n<li><span style=\"font-family: 'times new roman', times, serif\">If we take multivariable H-function in place of I-function in (2.1), then we have a known result obtained by \u00a0Chaurasia and Singhal [2] as following:<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1170\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/44.png\" alt=\"\" width=\"663\" height=\"536\" srcset=\"https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/44.png 663w, https:\/\/www.gyanvihar.org\/journals\/uploads\/2018\/12\/44-624x504.png 624w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0\u00a0<\/span><span style=\"font-family: 'times new roman', times, serif\">\u00a0Valid under the conditions surrounding (2.1)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">Taking R=1 and replacing f\u2192f<sub>1<\/sub>,\u2026.,f<sub>r<\/sub> in (2.1),<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">\u00a0we get a known result obtained by Chaurasia<\/span><span style=\"font-family: 'times new roman', times, serif\">\u00a0and Shekhawat [1].<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>Conclusion: <\/strong><\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">The main result derived here is of a very general\u00a0<\/span><span style=\"font-family: 'times new roman', times, serif\">Nature and hence encompass several cases of interest hitherto scattered in the literature.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\"><strong>REFERENCES<\/strong>:<\/span><\/p>\n<ol style=\"text-align: justify\">\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">B.L. Chaurasia and A.S. Shekhawat, fractional derivative associated with the multivariable polynomials, Kyungpook math. J. <strong>47 <\/strong>(2007),495-500.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">B.L. Chaurasia, fractional derivative of the multivariable polynomials, Bull. Malaysian Math. Sc. Soc.(second series), <strong>26<\/strong> (2003),1-8<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">B. Oldham and J. Spanier, The fractional calculus, Academic Press New York, 1974.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">P.Saxena, The I-function, Anamaya Publishers, New Delhi (2008)<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">Srivastava, H.M., A multilinear generating function for the Konhauser sets of bi-orthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. <strong>117<\/strong>, 183-191 (1985)<\/span><\/li>\n<li style=\"text-align: justify\">M. Srivastava, and M.C. Daoust, certain generalized neuman expansions associated with the kempe\u2019 de f\u2019eriet function, Nederl.Akad.Wetensch Indag.Math., <strong style=\"font-family: 'times new roman', times, serif\">31<\/strong><span style=\"font-family: 'times new roman', times, serif\"> (1969), 449-457.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">M. Srivastava, and M. Garg, some integrals involving a general class of polynomials and the multivariable H-function, Rev. Roumanie phys.,<strong>32<\/strong> (1987), 685-692<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">M. Srivastava, and S.P.Goyal, fractional derivatives of the H-function of several variables, J.Math. Anal. Appl. <strong>112<\/strong> (1985), 641-651.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"font-family: 'times new roman', times, serif\">M. Srivastava, and R. Panda, certain expansion formulas involving the generalized Lauricella function, II comment math. Univ. St. Paul, <strong>24<\/strong> (1974), 7-14<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify\">\n","protected":false},"excerpt":{"rendered":"<p>*HARSHITA GARG AND **ASHOK SINGH SHEKHAWAT *Suresh Gyan Vihar University, Jagatpura, Jaipur, Rajasthan, India **Arya College Of Engineering And Information Technology, Jaipur, Rajasthan, India \u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0Abstract : The object of this paper is to obtain a fractional derivative of I- function associated with generalize Lauricella\u00a0 functions and general \u00a0\u00a0class of multivariable polynomials. \u00a0Key words: [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,40],"tags":[],"class_list":["post-1118","post","type-post","status-publish","format-standard","hentry","category-journal-of-environment-science-and-technology","category-volume-1-issue-2-2015-journal-of-environment-science-and-technology"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>research journal - Research Journal<\/title>\n<meta name=\"description\" content=\"The paper deals to obtain a fractional derivative of I-function associated with generalized Lauricella functions &amp; general class of multivariable polynomial\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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