Abstract: This paper focuses on the interrelationship between beta, k-beta, and extended k-beta function of the matrix argument. The study highlights these function's mathematical properties, functional relations, integral formula, integral representation, and interrelationships, highlighting their applications and importance in the context of matrix argument.
Additionally, we aim to create a stronger relation that enables the derivation of further results applicable in fractional calculus. Here in this paper, we applied some basic properties of beta and gamma functions to obtain new identities and relationships. We also used Laplace transform and the Legendre-duplication formulas to get new relations of beta, k-beta, and extended k-beta functions of matrix arguments. Of particular interest is the use of this transformation to produce identities that offer improved solutions to challenging issues about the beta function. We hope that this article will be helpful to the new researchers for their further research work and applications of fractional calculus.