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Integral Representations and Numerical Evaluation of the Two-Variable H-Function.

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Title: Integral Representations and Numerical Evaluation of the Two-Variable H-Function.
Authors: Kumari, Ruby1 rubykumari9534@gmail.com, Ahmed, Lukman1 luqmanahmed513@gmail.com, Awasthi, A. K.2 dramitawasthi@gmail.com
Source: IAENG International Journal of Applied Mathematics. Dec2025, Vol. 55 Issue 12, p4152-4158. 7p.
Subjects: Integral representations, Mathematical functions, Field theory (Physics), Mathematical physics, Integrals, Applied mathematics, Statistical models
Abstract: This paper explores the intricate landscape of the H-function of two variables, extending the classical understanding of Fox's H-function. By delving into the double Barnes integral representation, this study introduces a new formulation and expansion of the H-function, catering to two variables with a comprehensive parameter set, by defining and representing the H-function of two variables through a double Barnes integral. This integral representation allows for a robust mathematical structure that can be utilized to tackle complex problems in applied mathematics and theoretical physics. By utilizing the work of Prasad and Gupta, we provide a detailed expansion that enhances the understanding of the Hfunction's behavior and its potential applications. This expansion is crucial for solving complex equations and modeling sophisticated systems in various scientific disciplines. We also derive a main integral involving the H-function of two variables, demonstrating its convergence and validity under specific conditions. These integrals play a vital role in simplifying and solving complex integrals encountered in various applications. Moreover, we explore several particular cases of our main integral, illustrating its versatility and broad applicability. By expanding the traditional H-function to two variables and offering novel formulations, expansion formulas, and integral representations, this study significantly advances the subject of applied mathematics. These developments open up new avenues for study and application in statistical distribution theory, quantum physics, partial differential equations, and other fields. [ABSTRACT FROM AUTHOR]
Copyright of IAENG International Journal of Applied Mathematics is the property of International Association of Engineers (IAENG) and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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  Data: Integral Representations and Numerical Evaluation of the Two-Variable H-Function.
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  Data: <searchLink fieldCode="AR" term="%22Kumari%2C+Ruby%22">Kumari, Ruby</searchLink><relatesTo>1</relatesTo><i> rubykumari9534@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Ahmed%2C+Lukman%22">Ahmed, Lukman</searchLink><relatesTo>1</relatesTo><i> luqmanahmed513@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Awasthi%2C+A%2E+K%2E%22">Awasthi, A. K.</searchLink><relatesTo>2</relatesTo><i> dramitawasthi@gmail.com</i>
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  Data: <searchLink fieldCode="JN" term="%22IAENG+International+Journal+of+Applied+Mathematics%22">IAENG International Journal of Applied Mathematics</searchLink>. Dec2025, Vol. 55 Issue 12, p4152-4158. 7p.
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  Data: <searchLink fieldCode="DE" term="%22Integral+representations%22">Integral representations</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+functions%22">Mathematical functions</searchLink><br /><searchLink fieldCode="DE" term="%22Field+theory+%28Physics%29%22">Field theory (Physics)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+physics%22">Mathematical physics</searchLink><br /><searchLink fieldCode="DE" term="%22Integrals%22">Integrals</searchLink><br /><searchLink fieldCode="DE" term="%22Applied+mathematics%22">Applied mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+models%22">Statistical models</searchLink>
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  Label: Abstract
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  Data: This paper explores the intricate landscape of the H-function of two variables, extending the classical understanding of Fox's H-function. By delving into the double Barnes integral representation, this study introduces a new formulation and expansion of the H-function, catering to two variables with a comprehensive parameter set, by defining and representing the H-function of two variables through a double Barnes integral. This integral representation allows for a robust mathematical structure that can be utilized to tackle complex problems in applied mathematics and theoretical physics. By utilizing the work of Prasad and Gupta, we provide a detailed expansion that enhances the understanding of the Hfunction's behavior and its potential applications. This expansion is crucial for solving complex equations and modeling sophisticated systems in various scientific disciplines. We also derive a main integral involving the H-function of two variables, demonstrating its convergence and validity under specific conditions. These integrals play a vital role in simplifying and solving complex integrals encountered in various applications. Moreover, we explore several particular cases of our main integral, illustrating its versatility and broad applicability. By expanding the traditional H-function to two variables and offering novel formulations, expansion formulas, and integral representations, this study significantly advances the subject of applied mathematics. These developments open up new avenues for study and application in statistical distribution theory, quantum physics, partial differential equations, and other fields. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of IAENG International Journal of Applied Mathematics is the property of International Association of Engineers (IAENG) and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Field theory (Physics)
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              Text: Dec2025
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