A Note on Some Reduction Formulas for the Incomplete Beta Function and the Lerch Transcendent

We derive new reduction formulas for the incomplete beta function Bν,0,z and the Lerch transcendent Φz,1,ν in terms of elementary functions when ν is rational and z is complex. As an application, we calculate some new integrals. Additionally, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.

Logarithmic concavity of the inverse incomplete beta function with respect to the first parameter

The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as a univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.

A faster and more accurate algorithm for calculating population genetics statistics requiring sums of Stirling numbers of the first kind

Stirling numbers of the first kind are used in the derivation of several population genetics statistics, which in turn are useful for testing evolutionary hypotheses directly from DNA sequences. Here, we explore the cumulative distribution function of these Stirling numbers, which enables a single direct estimate of the sum, using representations in terms of the incomplete beta function. This estimator enables an improved method for calculating an asymptotic estimate for one useful statistic, Fu’s Fs. By reducing the calculation from a sum of terms involving Stirling numbers to a single estimate, we simultaneously improve accuracy and dramatically increase speed.

DOUBLE SPEND RACES

We correct the double spend race analysis given in Nakamoto’s foundational Bitcoin article and find the exact closed-form formula for the probability of success of a double spend attack using the regularized incomplete beta function. We give the first proof of its exponential decay on the number of confirmations, often cited in the literature, and find an asymptotic formula. Larger number of confirmations are required compared to those given by Nakamoto. We also compute this probability conditional to the knowledge of the time of the confirmations. This provides a finer risk analysis than the classical one.

Some results on the beta function and the incomplete beta function

The incomplete Beta function [Formula: see text] is defined for [Formula: see text] and [Formula: see text]. This definition was extended to negative integer values of [Formula: see text] and [Formula: see text] by Özçaḡ et al. using neutrix limits. Partial derivatives of the incomplete Beta function [Formula: see text] for negative integer values of [Formula: see text] and [Formula: see text] were then evaluated. In the following, a number of results are obtained for the incomplete Beta function [Formula: see text] and then the corresponding results for the Beta function [Formula: see text] are deduced.