Ray trajectories, binomial of a new type, and the binary system; on binomial distribution of the second (nonlinear) type for big binomial power

TL;DR

This paper introduces a new nonlinear binomial distribution, explores its geometric interpretation through ray systems, and compares its properties with traditional binomial distributions for large powers.

Contribution

It presents a novel algorithm for constructing a nonlinear arithmetic triangle and defines a new type of binomial coefficients with geometric and numerical analysis.

Findings

Nonlinear binomial coefficients can be constructed via a new algorithm.

Envelopes of sums of nonlinear binomial coefficients are similar to classical cases at large powers.

A new empirical formula for the half-sums of nonlinear binomial coefficients is proposed.

Abstract

The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be binomial coefficients of a new type. An analogy has been drawn with the binomial coefficients calculated with the use of the Pascal triangle. The paper provides a geometrical interpretation of binomials of different types in considering the branching systems of rays. Results of numerical calculations of binomial distribution of the second (nonlinear) type for big power of a binomial are given. Difference of geometrical properties of linear and nonlinear arithmetic triangles and envelopes of binomial distributions of the first and second types is drawn. The empirical formula for half-sums of binomial coefficients of the second (nonlinear) type is offered. Comparison of envelopes of binomial coefficients sums is carried out. It is shown that at big degrees of a binomial a form of envelops of these sums are close.