Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions

Abstract

In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao geometry of the Dirichlet distributions in the two-dimensional case.

A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments

“…is completely monotonic with respect to the variable v ∈ (0, ∞). For details about completely monotonic functions, please refer to the review article [35] and closely related references therein.…”

Two Approximation Formulas for Gamma Function with Monotonic Remainders

“…The derivative of ln Γ(r), denoted by ψ(r) = Γ ′ (r) Γ(r) , is called the digamma function and the derivatives ψ (n) (r) for n ≥ 0 are referred to as the polygamma functions. For more information about gamma function and polygamma functions see [2][3][4][5][6], as well as the closely linked references therein, for further details.…”