Numerical and graphical exploration of the generalized beta-logarithmic matrix function and its properties

TL;DR

The paper introduces the generalized beta-logarithmic matrix function (GBLMF) by unifying the generalized beta matrix function with the logarithmic mean through a two-parameter Mittag-Leffler matrix framework. It defines both the generalized beta matrix function $\mathfrak{B}_{(\phi,\psi)}^{(R,S,\eta,\xi)}(P,Q)$ and the generalized beta-logarithmic matrix function $\mathfrak{BL}_{(\phi,\psi)}^{(R,S,\eta,\xi)}(a,b;P,Q)$, and shows how classical ($\mathfrak{B}$) and extended ($\mathfrak{B}^{R}$) beta matrix functions are recovered as special cases. The authors derive key properties, including symmetry, scaling, and addition relations, along with infinite/finite sum representations, integral representations, and higher-order derivative formulas, establishing a comprehensive theoretical framework. Numerical examples and graphical analyses illustrate the behavior of the new function and highlight its advantages over classical and earlier beta matrix functions, particularly in terms of flexibility and potential convergence properties. The work sets the stage for future exploration of matrix inequalities, numerical methods, and diverse applications in matrix analysis and related fields.

Abstract

This paper investigates the generalized beta-logarithmic matrix function (GBLMF),which combines the extended beta matrix function and the logarithmic mean. The study establishes essential properties of this function, including functional relations, inequalities, finite and infinite sums, integral representations, and partial derivative formulas. Theoretical results are accompanied by numerical examples and graphical representations to demonstrate the behavior of the new matrix function. Additionally, a comparison with classical and previously studied beta matrix functions is presented to highlight the differences and advantages of the generalized version. The findings offer valuable insights into the properties and applications of the extended beta-logarithmic matrix function in various mathematical and applied contexts.

Figures (8)

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Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• proof
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof