Advanced Theorems on Double Integral Transforms and Their Applications

This article delves into the realm of double integral transforms (DIT), focusing on the pivotal role played by Fox's H-Function in their theoretical framework. The DIT denoted as <img src=image/13439778_01.gif> is intricately defined through Fox's HFunction and expressed as a Mellin-Barnes-type contour integral with various parameters and conditions. The study emphasizes chain properties connecting the DIT, presenting a concise representation as <img src=image/13439778_02.gif>. Three theorems are established, leveraging the power series expansions of special functions such as Laplace transforms, Hankel transforms, and specific transforms by Pathak and Narain. These theorems are proven analytically and their results are verified with the help of examples. The Fox H-Function is a powerful mathematical tool, which is explored for its significance in the analysis of DIT. Being a generalization of the hypergeometric series, the H-function finds widespread applications across mathematics, physics, and engineering. Detailed proofs substantiate the three theorems, illustrating the manipulation of the double integral transform under specific conditions. The application of these theorems extends to the evaluation of known and novel integrals involving the product of H-Function and other mathematical functions. This comprehensive exploration unveils the indispensable role of Fox's H-Function in the theoretical landscape of double integral transform and its applications.