Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Integral Transform and its Multi-dimensional Analog with their Various Specialized Forms
Ahmed, L., Kumari, R., and Awasthi, A. K.
Corresponding Email: dramitawasthi@gmail.com
Received date: 6 February 2025
Accepted date: 3 July 2025
Abstract:
This paper presents advanced techniques for transforming integrals involving special functions, focusing on extending integral transformations to multi$-$dimensional cases. Building on classical methods such as Euler and Laplace transformations, the study introduces new triple integral transformations, which are then generalized to
higher dimensions. The key contribution of the research is the derivation of triple integral transformations for generalized hypergeometric functions and special polynomials like the Bessel and Rice polynomials. These transformations reduce complex integrals to more manageable forms, utilizing Gamma functions and hypergeometric series. The paper also introduces multi$-$dimensional analogs of these transformations, allowing the techniques to be applied to integrals with multiple variables common in mathematical physics and engineering problems. A notable extension is the incorporation of Fox$'$s H$-$function, which broadens the applicability of the transformations to more generalized hypergeometric functions. This enables solutions to a wider range of
mathematical problems, particularly complex integrals. Additionally, the paper presents various specialized forms of transformations, including generating relations, expansion, and summation formulas for well-known polynomials such as Gegenbauer and Bessel. The operational techniques offered a systematic framework for handling complex multidimensional integrals, making the methods applicable across theoretical and applied mathematics. This work provides a significant advancement in integral transform theory in mathematical analysis, physics, and engineering.
Keywords: triple integral transformation; generalized hypergeometric function; integral transforms; Gegenbauer polynomial
