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Motivated by applications in Bayesian analysis, we introduce a multidimensional beta distribution in the ordered simplex. We study the properties of this distribution and connect them with the generalized incomplete beta function. This function is crucial in applications of multidimensional beta distribution. Thus, we present two efficient numerical algorithms for computing the generalized incomplete beta function, one based on Taylor series expansion and another based on Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

In this article, we aim to present a new extension of the beta-logarithmic function with the help of an extended beta function and logarithmic mean. Then we introduce its statistical properties and investigate some basic important formulas of extended beta-logarithmic functions, such as integral representation, summation formula and Mellin transform. Further, we consider a new type of hypergeometric and confluent hypergeometric functions using this extended beta-logarithmic function and discuss their various result. [ABSTRACT FROM AUTHOR]