reapress null reapress3042-30903042-3090 reapress https://doi.org/10.22105/kmisj.vi.144 Research Article Special functions, Gamma function, Factorial generalization, Beta functions. An Introductory Overview of The k−Analogue of Gamma Functions and Related FunctionsAn Introductory Overview of The k−Analogue of Gamma Functions and Related Functions Abdalla Mohamed Department of Mathematics, College of Science, Khalid University, AlQura’a, Abha, P.O. Box 960, Saudi Arabia. Amani Saad Alasmry Amani Saad Department of Mathematics, College of Science, Khalid University, AlQura’a, Abha, P.O. Box 960, Saudi Arabia. Alqarni Mohammed Z. Department of Mathematics, College of Science, Khalid University, AlQura’a, Abha, P.O. Box 960, Saudi Arabia. 09 2026 04 09 2026 3 3 © 2026 reapress 2026

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 An Introductory Overview of The k−Analogue of Gamma Functions and Related Functions

The Gamma function extends the factorial to a continuous and complex-variable setting and provides a foundation for numerous special functions. This paper examines its basic properties, related functions, and incomplete forms before investigating selected applications to definite integration, statistical mechanics, probability theory, and fractional calculus. Particular attention is given to the conditions under which integral representations and identities are valid. Classical results are connected with recent work on series representations and Stirling-type approximations. The study emphasizes the gamma function both as an analytical object and as a unifying tool in mathematical physics.

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