Explicit evaluations of subfamilies of the hypergeometric function $ _3F_2(1) $ along with specific fractional integrals

Abdelhamid Zaidi; Saleh Almuthaybiri; Abdelhamid Zaidi; Saleh Almuthaybiri

doi:10.3934/math.2025264

AIMS Mathematics

2025, Volume 10, Issue 3: 5731-5761. doi: 10.3934/math.2025264

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Research article

Explicit evaluations of subfamilies of the hypergeometric function $ _3F_2(1) $ along with specific fractional integrals

Abdelhamid Zaidi ,
Saleh Almuthaybiri ^,

Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia

Received: 02 February 2025 Revised: 01 March 2025 Accepted: 07 March 2025 Published: 14 March 2025
MSC : 26A33, 33C05, 33C20

The present study explores the application of hypergeometric functions in evaluating fractional integrals, providing a comprehensive framework to bridge fractional calculus and special functions. As a generalization of classical integrals, fractional integrals have gained prominence due to their wide applicability in modeling anomalous diffusion, viscoelastic systems, and other non-local phenomena. Hypergeometric functions, renowned for their rich analytical properties and ability to represent solutions to differential equations, offer an elegant and versatile tool for solving fractional integrals. In this paper, we evaluate a new class of fractional integrals, presenting results that contribute significantly to the study of generalized hypergeometric functions, particularly $ _3F_2(1) $. The results reveal previously unexplored connections within these functions, providing new insights and extending their applicability. Furthermore, evaluating these fractional integrals holds promise for advancing the theoretical understanding and practical applications of fractional differential equations.
- fractional integrals,
- fractional derivatives,
- Gauss hypergeometric functions,
- generalized hypergeometric functions,
- numerical series,
- power series,
- decomposition into partial fractions,
- differential equations
Citation: Abdelhamid Zaidi, Saleh Almuthaybiri. Explicit evaluations of subfamilies of the hypergeometric function $ _3F_2(1) $ along with specific fractional integrals[J]. AIMS Mathematics, 2025, 10(3): 5731-5761. doi: 10.3934/math.2025264

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Abstract

The present study explores the application of hypergeometric functions in evaluating fractional integrals, providing a comprehensive framework to bridge fractional calculus and special functions. As a generalization of classical integrals, fractional integrals have gained prominence due to their wide applicability in modeling anomalous diffusion, viscoelastic systems, and other non-local phenomena. Hypergeometric functions, renowned for their rich analytical properties and ability to represent solutions to differential equations, offer an elegant and versatile tool for solving fractional integrals. In this paper, we evaluate a new class of fractional integrals, presenting results that contribute significantly to the study of generalized hypergeometric functions, particularly $ _3F_2(1) $. The results reveal previously unexplored connections within these functions, providing new insights and extending their applicability. Furthermore, evaluating these fractional integrals holds promise for advancing the theoretical understanding and practical applications of fractional differential equations.

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