Exploring the Extended Beta-Logarithmic Function: Matrix Arguments and Properties

: The beta-logarithmic function substantially generalizes the standard beta function, which is widely recognized for its significance in many applications. This article is devoted to the study of a generalization of the classical beta-logarithmic function in a matrix setting called the extended beta-logarithmic matrix function. The proofs of some essential properties of this extension, such as convergence, partial derivative formulas, functional relations, integral representations, inequalities, and finite and infinite sums, are established. Moreover, an application of the extended beta-logarithmic function in matrix arguments is proposed in probability theory. Further, numerical examples and graphical presentations of the new generalization are obtained


Introduction and Preliminaries
Special functions, which are natural generalizations of elementary functions, are derived by solving partial differential equations that satisfy a specific set of conditions.For centuries, many special functions have been developed and implemented in various fields, including combinatorics, chemistry, statistics, physics, and engineering (see, e.g., [1][2][3]).
On the other hand, recent extensions of special functions build upon the work of esteemed researchers such as Abdalla et al. [4,5], Abd-Elmageed et al. [6], Hidan et al. [7], Fuli He et al. [8], and Cuchta et al. [9], who have shown a strong interest in studying the extension of special functions in matrix arguments.Contributions to the field by considering various extensions of the gamma, beta, and hypergeometric matrix functions have been documented in [10][11][12][13][14]. Inspired by earlier studies, a new extension of the beta function in its matrix version is presented: the extended beta-logarithmic matrix function (EBLMF).An application and discussion of some of its analytic and numerical properties were also provided.One can easily obtain various applications from the close relationship of EBLMF with several special functions.
This paper is structured as follows: Section 2 gives the definition of the extended beta-logarithmic matrix function and studies its convergence properties and partial derivative formulas.Section 3 presents various properties of the EBLMF, including functional relations, inequalities, infinite sums, finite sums, and integral formulas.In Section 4, an application of the extended beta-logarithmic function in matrix arguments is offered.Section 5 provides numerical illustration examples and graphical descriptions of the EBLMF and some exceptional cases.Finally, in Section 6, we conclude with some final remarks.
Remark 1.One can deduce some special cases of Definition (1) as follows: 17) reduces to the extended beta matrix function defined in (11).
b-When T = 0 κ in (17), a new matrix version of the beta-logarithmic function ( 14) is brought in as where 0 κ denotes the zero matrix of any order κ.
c-To attain the beta matrix function defined in (9), Then, the extended logarithmic mean of a matrix argument is defined by e-When T = 0 κ in (19), one obtains a scalar logarithmic mean given in (12).

Partial Derivative Formulas
Here, the higher-order derivative formulas of EBLMF where the matrices Λ 1 , Λ 2 , T ∈ M κ and the parameters α, β ≥ 0 are discussed in the following theorem.
Setting T = 0 κ in Theorem 1, the higher-order derivative formulas for the betalogarithmic matrix function defined in (18) are given in the following corollary. and Similarly, the following result gives the partial differentiations of the logarithmic function in a matrix argument (19). and Remark 3. When considering the results in Remark 1 and Theorem 1, various other outcomes in the literature, including those in (see, e.g., [20,21,25]), can be demonstrated as special cases.

Some Analytic Characteristics
The essential analytic properties are established in this section.
Theorem 2. The EBLMF defined in (17) satisfies the following functional relation: Proof.Using definition 1, we obtain Thus, the desired result is attained.
Proof.From ( 11) and ( 13), we obtain After applying Young's inequality and simplifying, we find We can derive the desired assertion in (39) by making use of (40) and (41).
Corollary 5.For T = 0 κ in Theorem 3, the following inequality for BL α,β [Λ 1 , Λ 2 ] holds true: and T be reciprocally commutative matrices in M κ , we derive The following theorem gives various integral representations of the EBLMF defined in (17).
Corollary 8.For α, β ∈ R + such that α ̸ = β and let T ∈ M κ , we have Remark 5.There are also matrix versions of some known infinite sums in [25].

The Beta-Logarithmic Distribution: Matrix Arguments
As is well known, the beta distribution is one of the essential variate distributions in statistical analysis, and its importance is that it is in deriving moments and can help us understand the different probabilities associated with the random variable X (cf.[29]).Traditional beta distributions were introduced in [11][12][13]28,30] using extended beta functions.They suggested that these distributions could help analyze and review techniques employed in specific circumstances during project evaluation and review.They pointed out that these distributions could be advantageous for evaluating and reviewing the method used in particular cases during project evaluation and review.Here, the generalized beta-logarithmic distribution of matrix arguments is defined as where α, β ∈ R + with α ̸ = β and let Λ 1 , Λ 2 , T ∈ M κ .It will be said that a random variable X with probability density matrix function defined by (54) has the extended betalogarithmic distribution with matrix arguments Λ 1 , Λ 2 , and T in M κ .If the incomplete extended beta-logarithmic matrix function is expressed by then the cumulative distribution of (54) can be given as Remark 6.The incomplete generalized beta-logarithmic matrix function in (55) can be reduced to numerous simple incomplete extended beta matrix functions (see, e.g., [4,13]).
Also, let ℘ ∈ R + , then one obtains (cf.[11,28]) Following the particular case of (57) at ℘ = 1, the mean of the distribution is provided as Further, the variance of the generalized beta-logarithmic distribution of matrix arguments is shown as In addition, the moment-generating function of the distribution can be expressed as Remark 7. The results in [11] can be achieved when α = β = 1 in the above results.Also, setting α = β = 1, in (54), corresponding to (g) in Remark 2.1, one can achieve various results in [28] (Chapter 5, p.258).
Remark 8.The generalized results of the matrix setting in [25] (p.137) are obtained by taking T = 0 κ in the above results.Thus, we have

Example 4. For
Then, Thus, we obtain Thus, we obtain Thus, we find that (66)

Graphical Representations
This section illustrates the generalization presented in this paper with the previous results in [20] in their graphical forms to provide a comprehensive understanding.The differences were handled using the infinite norm (3) since the comparisons run over matrices.The size of all matrices used here is 2 × 2. The choice of the used matrices , as in [20], while T is a matrix chosen to be approaching the zero matrices, 0 κ , κ = 2.
Figure 1 shows the graph of the differences between the EBLMF and beta matrix function against various values of α in the interval (0, 20) for multiple choice of T matrix having infinity-norm starting from 0.5 and approaching 0 and taking β = 1.As α ′ s value increases, the difference decreases when T is not close to 0 κ .However, for T approaching zero matrices, the difference decreases for small values of α, and then it changes direction.That is because of the trade-off between α and the choice of T in (17).Similar observations can be made when fixing α = 1 and plotting the differences versus β ∈ (0, 20), as shown in Figure 2.
When the matrix T approaches zero, ∥EBLMF − B(Λ 1 , Λ 2 )∥ ∞ should go to zero when α, β = 1, and the result in [20] is obtained.To show the generalization introduced in this paper, Figure 3 depicts the graphs of the differences ∥EBLMF − B(Λ 1 , Λ 2 )∥ ∞ versus ∥T − 0 κ ∥ ∞ when the matrix T is chosen to approach zero, and the values of α and β = 0.2, 0.4, 0.6.One can conclude that the differences decrease for all the values of α and β when T tends to zero in its infinity norm.Similar results can be obtained from Figure 4 when the values of α and β = 0.5, 1.5, 3 for a suitable choice of T matrix.

Conclusions
Later, many researchers contemplated the extension of the classical beta function to the matrix framework.The extended beta-logarithmic matrix function extends the extended beta matrix function [10] and the beta matrix function [20].This manuscript explores several analytical properties of this function and employs this extension to derive the generalized beta distribution of matrix arguments.In addition, some numerical illustration examples and graphical descriptions were presented to show the efficacy of this extension by using MATLAB R2023b.Finally, the extended beta-logarithmic matrix function has the potential to extend several known applications based on numerous special functions in the literature.

Funding:
This work was funded by the Deanship of Scientific Research at King Khalid University through a large group research project under grant number RGP2/327/45.Data Availability Statement: No data were used to support this study.
For α = 2, β = 4, Λ 1 = The previous examples show that if Λ 1 , Λ 2 , and T do not commute with α ̸ = β, then the property symmetry of the function B T