{"id":13988,"date":"2025-07-10T20:17:14","date_gmt":"2025-07-10T19:17:14","guid":{"rendered":"https:\/\/icertpublication.com\/?page_id=13988"},"modified":"2026-04-03T10:23:22","modified_gmt":"2026-04-03T04:53:22","slug":"a-study-of-generalized-hypergeometric-function-and-its-applications-in-vary-disciplines","status":"publish","type":"post","link":"https:\/\/icert.org.in\/index.php\/2025\/07\/10\/a-study-of-generalized-hypergeometric-function-and-its-applications-in-vary-disciplines\/","title":{"rendered":"A Study of Generalized Hypergeometric Function and its Applications in Vary Disciplines"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"13988\" class=\"elementor elementor-13988\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d29049b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d29049b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5f5799a\" data-id=\"5f5799a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6787ae4 elementor-widget elementor-widget-heading\" data-id=\"6787ae4\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Shodh Sari-An International Multidisciplinary Journal<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-88d59af elementor-widget elementor-widget-heading\" data-id=\"88d59af\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Vol-04, Issue-03(Jul-Sep  2025)<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6383078 elementor-widget elementor-widget-heading\" data-id=\"6383078\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\">An International scholarly\/ academic journal, peer-reviewed\/ refereed journal, ISSN : 2959-1376<\/h5>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e74f2f5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e74f2f5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-093dfbc\" data-id=\"093dfbc\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4a84514 elementor-widget elementor-widget-heading\" data-id=\"4a84514\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">A Study of Generalized Hypergeometric Function and its Applications in\nVary Disciplines<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9df6432 elementor-widget elementor-widget-text-editor\" data-id=\"9df6432\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: center; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mahala, Mukesh Kumar<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"><span style=\"font-size: 0.6em; vertical-align: super;\">1 <\/span><\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">and Sharma, Rahul<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"><span style=\"font-size: 0.6em; vertical-align: super;\">2<\/span><\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: center; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"><span style=\"font-size: 0.6em; vertical-align: super;\">1<\/span><\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Department of Mathematics, University of Engineering &amp; Management Jaipur, Jaipur, Rajasthan, India<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: center; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; background-color: transparent; font-family: Cambria, serif; color: #000000; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-variant-position: normal; font-variant-emoji: normal; vertical-align: baseline; white-space-collapse: preserve;\"><span style=\"font-size: 0.6em; vertical-align: super;\">2<\/span><\/span><span style=\"font-size: 12pt; background-color: transparent; font-family: Cambria, serif; color: #000000; font-variant-numeric: normal; font-variant-east-asian: normal; font-variant-alternates: normal; font-variant-position: normal; font-variant-emoji: normal; vertical-align: baseline; white-space-collapse: preserve;\">Department of Mathematics, Seth Gyaniram Bansidhar Podar College, Nawalgarh, Rajasthan, India<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4bf8f1f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4bf8f1f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f14d7b6\" data-id=\"f14d7b6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-aa9b0a0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"aa9b0a0\" data-element_type=\"section\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-d270afb\" data-id=\"d270afb\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e105ff8 elementor-widget__width-initial elementor-widget elementor-widget-text-editor\" data-id=\"e105ff8\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">DOI: <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/www.google.com\/search?q=https:\/\/doi.org\/10.59231\/SARI7836\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #0563c1; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.59231\/SARI7836<\/span><\/a><\/p><p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Subject: Mathematics \/ Applied Sciences<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Page No: 46-56<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-2cab604\" data-id=\"2cab604\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f271234 elementor-widget__width-initial elementor-widget elementor-widget-text-editor\" data-id=\"f271234\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Received on March 05, 2025\u00a0<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Accepted on May 10, 2025\u00a0<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Published on July 01, 2025<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-113a56a elementor-widget__width-initial elementor-widget elementor-widget-text-editor\" data-id=\"113a56a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.2; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: Calibri,sans-serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Thematic Classification: Generalized Hypergeometric Function, Mathematical Modeling, Applied Mathematics, Computational Functions, Interdisciplinary Applications.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-78bea5f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"78bea5f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-63a7be0\" data-id=\"63a7be0\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1e2af40 elementor-widget elementor-widget-heading\" data-id=\"1e2af40\" data-element_type=\"widget\" id=\"Shodh-Sari-v4-i3-4A\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h6 class=\"elementor-heading-title elementor-size-default\">Abstract\n<\/h6>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d90e7d4 elementor-widget elementor-widget-text-editor\" data-id=\"d90e7d4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Elementary functions, Bessel functions, Legendre functions and many other special functions are included in the large family of mathematical functions known as generalized hypergeometric functions. A power series with coefficients that are rational functions of the index defines them. They are used in many disciplines, such as engineering, statistics and physics, because of their rich mathematical features and adaptability. The beauty and interdependence of mathematical ideas are demonstrated by the Generalized Hypergeometric Function. Researchers and practitioners from a wide range of disciplines find it to be an indispensable tool due to its unifying power, rich analytical features, and broad applications. Numerous unusual functions are included as particular examples of the generalized Hypergeometric function. Legendre polynomials, Bessel functions, the confluent Hypergeometric function, and numerous more noteworthy examples are also included. An order <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(q+1)<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> linear homogeneous differential equation is satisfied by the generalized hypergeometric function. In many applications, but especially in mathematical physics, this differential equation is essential. It is possible to write the generalized hypergeometric function in terms of contour integrals, which offers different representations and makes it easier to evaluate some integrals. The generalized Hypergeometric function has a wealth of transformation formulas that allow one Hypergeometric function to be transformed into another with distinct parameters. These transformations are quite useful for examining relationships between various special functions and simplifying expressions.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Keywords<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">: Special Function, Generalized Hypergeometric functions, Recurrence Relations<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f0ad94a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f0ad94a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e87ffed\" data-id=\"e87ffed\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3054d45 elementor-widget elementor-widget-heading\" data-id=\"3054d45\" data-element_type=\"widget\" id=\"Shodh-Sari-v4-i3-4i\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h6 class=\"elementor-heading-title elementor-size-default\">Impact Statement\n<\/h6>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-61829d7 elementor-widget elementor-widget-text-editor\" data-id=\"61829d7\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">This study investigates the generalized hypergeometric function, a powerful and unifying mathematical tool that encompasses a wide range of special and elementary functions. By establishing novel representations and transformation identities, the work enhances understanding of the hypergeometric function\u2019s role in solving complex differential equations, particularly those arising in physical models and computational mathematics. The ability to express numerous classical functions\u2014like Legendre, Bessel, and trigonometric functions\u2014as special cases of the generalized hypergeometric function provides a cohesive framework for theoretical and applied investigations. This contributes to both the simplification of mathematical modelling and the development of efficient computational methods.\u00a0<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d711eb0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d711eb0\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b037cb6\" data-id=\"b037cb6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-72add22 elementor-widget elementor-widget-heading\" data-id=\"72add22\" data-element_type=\"widget\" id=\"Shodh-Sari-v4-i3-4Aa\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h6 class=\"elementor-heading-title elementor-size-default\">About Author\n<\/h6>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5018992 elementor-widget elementor-widget-text-editor\" data-id=\"5018992\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mr. Mukesh Kumar Mahala is a dedicated academic professional with over 9 years of teaching experience in mathematics at the postgraduate level. He is currently serving as an Assistant Professor at Shri Shraddhanath P.G. College, Jhunjhunu. He has previously held similar roles at Shekhawati P.G. College, Sikar, and Shri Narayan P.G. College, Jaipur. He holds an M.Sc. in Mathematics and a B.Ed. and is presently pursuing a PhD from the University of Engineering &amp; Management, Jaipur.<\/span><\/p><p><b id=\"docs-internal-guid-b4d9f312-7fff-985a-3f5f-cf131943d25b\" style=\"font-weight: normal;\">\u00a0<\/b><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Dr. Rahul Sharma is currently serving as Associate Professor and Head of the Department of Mathematics at the University of Engineering &amp; Management, Jaipur, with over 13 years of teaching and research experience. He holds a Ph.D. in Mathematics from Amity University Rajasthan, specializing in special functions. Dr. Sharma has published extensively in Scopus and ESCI-indexed journals and actively contributes to academic conferences.\u00a0<\/span><\/p><p>\u00a0<\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Dr. Vidyadhar Sharma&#8217;s primary research interests lie in the fields of Applied &amp; Pure Mathematics, with a specific focus on Fractional Calculus and Special Functions. His work includes studies on subclasses of analytic functions, bi-univalent functions, meromorphic functions, and applications of fractional integral operators.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c52c3de elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c52c3de\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1f445ff\" data-id=\"1f445ff\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2366ba7 elementor-widget elementor-widget-heading\" data-id=\"2366ba7\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h6 class=\"elementor-heading-title elementor-size-default\">Cite this Article\n<\/h6>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a3d3af4 elementor-widget elementor-widget-text-editor\" data-id=\"a3d3af4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">APA 7th Style<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">\u00a0<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mahala, M. K., &amp; Sharma, R. (2025). A study of generalized hypergeometric function and its applications in vary disciplines. <\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Shodh Sari-An International Multidisciplinary Journal<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">4<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(03), 46\u201356. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.59231\/SARI7836\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #0563c1; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.59231\/SARI7836<\/span><\/a><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Chicago 17th Style<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">\u00a0<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mahala, Mukesh Kumar, and Rahul Sharma. &#8220;A Study of Generalized Hypergeometric Function and Its Applications in Vary Disciplines.&#8221; <\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Shodh Sari-An International Multidisciplinary Journal<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> 4, no. 3 (2025): 46\u201356. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.59231\/SARI7836\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #0563c1; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.59231\/SARI7836<\/span><\/a><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">MLA 9th Style<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">\u00a0<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mahala, Mukesh Kumar and Sharma, Rahul. &#8220;A Study of Generalized Hypergeometric Function and Its Applications in Vary Disciplines.&#8221; <\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Shodh Sari-An International Multidisciplinary Journal<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, vol. 4, no. 3, 2025, pp. 46-56, <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.59231\/SARI7836\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #0563c1; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.59231\/SARI7836<\/span><\/a><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e80a492 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e80a492\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c2ce653\" data-id=\"c2ce653\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e5abe21 elementor-widget elementor-widget-heading\" data-id=\"e5abe21\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h6 class=\"elementor-heading-title elementor-size-default\">Statements and Declarations\n<\/h6>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c311fd4 elementor-widget elementor-widget-text-editor\" data-id=\"c311fd4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Peer-Review Method:<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> This mathematical study underwent a double-blind peer-review process by two independent external experts in Mathematical Analysis and Applied Mathematics to ensure technical accuracy and theoretical rigor.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Competing Interests:<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> The authors (Mukesh Kumar Mahala and Rahul Sharma) declare that they have no financial or personal conflicts of interest that could have inappropriately influenced the research findings or mathematical derivations presented in this study.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Funding:<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> This research was conducted as part of the authors&#8217; scholarly activities at the University of Engineering &amp; Management Jaipur and Seth Gyaniram Bansidhar Podar College. No specific external grants or commercial funding were received for this work.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Data Availability:<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> The study is based on theoretical mathematical derivations and a review of existing literature on hypergeometric functions. All formulas and analytical data used are contained within the article or available through standard mathematical archives.<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Licence:<\/span> <span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">A Study of Generalized Hypergeometric Function and its Applications in Vary Disciplines<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> \u00a9 2025 by Mahala, M. K. and Sharma, R. is licensed under <\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">CC BY-NC-ND 4.0<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">. This work is published by the International Council for Education Research and Training (ICERT).<\/span><\/p><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 8pt;\"><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: bold; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Ethics Approval:<\/span><span style=\"font-size: 12pt; font-family: 'Times New Roman',serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> As this research consists of a theoretical mathematical analysis and does not involve human participants or animal subjects, it was deemed exempt from formal ethical review by the Institutional Research Committees of the authors&#8217; respective affiliations.<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4b1805d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4b1805d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2b4f723\" data-id=\"2b4f723\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-267a1e8 elementor-widget elementor-widget-heading\" data-id=\"267a1e8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h6 class=\"elementor-heading-title elementor-size-default\">References<\/h6>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2f35a24 elementor-widget elementor-widget-text-editor\" data-id=\"2f35a24\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol style=\"margin-top: 0; margin-bottom: 0; padding-inline-start: 48px;\"><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Agarwal, P. (2014). Certain properties of the generalized Gauss hypergeometric functions. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Applied Mathematics and Information Sciences<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">8<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(5)(5), 2315\u20132320. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.12785\/amis\/080526\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.12785\/amis\/080526<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Andrews, G. E. Bailey\u2019s transform, lemma, chains and tree, Special Functions 2000: Current Perspective and Future Directions, NATO Sci, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">II<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">: <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Math. Phys. Chem.<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\"> 30. Springer. (2001), 1\u201322.<\/span><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Andrews, G. E., Askey, R., &amp; Roy, R. (1999). Special functions, encyclopedia. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mathematica Applicata<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">. Cambridge University, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">71<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">.<\/span><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Arfken, G. (1985). <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Mathematical methods for physicists<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">. Academic Press<\/span><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Arshad, M., Choi, J., Mubeen, S., Nisar, K. S., &amp; Rahman, G. (2018). A new extension of the Mittag\u2013Leffler function. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Communications of the Korean Mathematical Society<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">33<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(2), 549\u2013560.<\/span><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Arshad, M., Mubeen, S., Nisar, K. S., &amp; Rahman, G. (2018). Extended Wright\u2013Bessel function and its properties. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Communications of the Korean Mathematical Society<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">33<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(1), 143\u2013155.<\/span><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Aslam Chaudhry, M. A., Qadir, A., Rafique, M., &amp; Zubair, S. M. (1997). Extension of Euler\u2019s beta function. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Journal of Computational and Applied Mathematics<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">78<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(1)(1), 19\u201332. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1016\/S0377-0427(96)00102-1\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1016\/S0377-0427(96)00102-1<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Brychkov, Y. A. (2008). <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Handbook of special functions: Derivatives, integrals, series and other formulas<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">. CRC Press. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1201\/9781584889571\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1201\/9781584889571<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Chaudhry, M. A., Qadir, A., Srivastava, H. M., &amp; Paris, R. B. (2004). Extended hypergeometric and confluent hypergeometric functions. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Applied Mathematics and Computation<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">159<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(2)(2), 589\u2013602. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1016\/j.amc.2003.09.017\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1016\/j.amc.2003.09.017<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Chaudhry, M. A., &amp; Zubair, S. M. (1994). Generalized incomplete gamma functions with applications. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Journal of Computational and Applied Mathematics<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">55<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(1)(1), 99\u2013124. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1016\/0377-0427(94)90187-2\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1016\/0377-0427(94)90187-2<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Kumar, S., &amp; Simran, S. (2024). Psychological impact of physical distancing due to covid 19 pandemic on school and higher education students. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Edumania-An International Multidisciplinary Journal<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">02<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(04), 101\u2013112. https:\/\/doi.org\/10.59231\/edumania\/9076<\/span><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Chaudhry, M. A., &amp; Zubair, S. M. (2001). <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">On a class of incomplete gamma functions with applications<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">. Chapman &amp; Hall\/CRC. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1201\/9781420036046\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1201\/9781420036046<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Choi, J., Rathie, A. K., &amp; Parmar, R. K. (2014). Extension of extended beta, hypergeometric and confluent hypergeometric functions. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Honam Mathematical Journal<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">36<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(2)(2), 357\u2013385. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.5831\/HMJ.2014.36.2.357\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.5831\/HMJ.2014.36.2.357<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Dragovic, V. (2002). The Appell hypergeometric functions and classical separable mechanical systems. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Journal of Physics A<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">35<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(9)(9), 2213\u20132221. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1088\/0305-4470\/35\/9\/311\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1088\/0305-4470\/35\/9\/311<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Galu\u00e9, L., Al-Zamel, A., &amp; Kalla, S. L. (2003). Further results on generalized hypergeometric functions. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Applied Mathematics and Computation<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">, <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">136<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">(1)(1), 17\u201325. <\/span><a style=\"text-decoration: none;\" href=\"https:\/\/doi.org\/10.1016\/S0096-3003(02)00014-0\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #0000ff; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: underline; -webkit-text-decoration-skip: none; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;\">https:\/\/doi.org\/10.1016\/S0096-3003(02)00014-0<\/span><\/a><\/p><\/li><li dir=\"ltr\" style=\"list-style-type: decimal; font-size: 11pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre; margin-left: -18pt; padding-left: 3.25pt;\" aria-level=\"1\"><p dir=\"ltr\" style=\"line-height: 1.7999999999999998; text-align: justify; margin-top: 0pt; margin-bottom: 0pt;\" role=\"presentation\"><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Koekoek, R., Lesky, P. A., &amp; Swarttouw, R. F. (2010). Hypergeometric orthogonal polynomials and their\u00a0<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">q<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">-analogues. <\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: italic; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">Springer Monographs in Mathematics<\/span><span style=\"font-size: 12pt; font-family: Cambria,serif; color: #000000; background-color: transparent; font-weight: 400; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;\">. Springer.<\/span><\/p><\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e059f98 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e059f98\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-07746b6\" data-id=\"07746b6\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7940a69 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7940a69\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-785bd16\" data-id=\"785bd16\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b3b1fc5 elementor-align-right elementor-widget__width-initial elementor-widget elementor-widget-button\" data-id=\"b3b1fc5\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/icert.org.in\/index.php\/shodh-sari-2\/hodh-sari-an-international-multidisciplinary-journal-vol-04-issue-02-apr-jun-2025\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Back<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Shodh Sari-An International Multidisciplinary Journal Vol-04, Issue-03(Jul-Sep 2025) An International scholarly\/ academic journal, peer-reviewed\/ refereed journal, ISSN : 2959-1376 A Study of Generalized Hypergeometric Function and its Applications in Vary Disciplines Mahala, Mukesh Kumar1 and Sharma, Rahul2 1Department of Mathematics, University of Engineering &amp; Management Jaipur, Jaipur, Rajasthan, India 2Department of Mathematics, Seth Gyaniram Bansidhar [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"site-sidebar-layout":"no-sidebar","site-content-layout":"page-builder","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"disabled","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[32],"tags":[],"class_list":["post-13988","post","type-post","status-publish","format-standard","hentry","category-shodh-sari-vol-04-issue-03-jul-sep-2025"],"acf":[],"_links":{"self":[{"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/posts\/13988","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/comments?post=13988"}],"version-history":[{"count":12,"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/posts\/13988\/revisions"}],"predecessor-version":[{"id":31878,"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/posts\/13988\/revisions\/31878"}],"wp:attachment":[{"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/media?parent=13988"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/categories?post=13988"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/icert.org.in\/index.php\/wp-json\/wp\/v2\/tags?post=13988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}