Diffusion of a particle via a stochastic process on the Pascal'e pyramid

TL;DR

This paper introduces a 3D generalization of Pascal's triangle called Pascal's cube, maps it onto the plane for easier computation, and models a stochastic process that obeys the heat equation.

Contribution

It constructs Pascal's cube, a 3D extension of Pascal's triangle, and demonstrates that a stochastic process on it satisfies the heat equation.

Findings

Pascal's cube can be mapped onto the plane for computation.

A stochastic process on Pascal's pyramid satisfies the heat equation.

The construction provides a new framework for modeling dispersion processes.

Abstract

In this note, we construct a $3$-dimensional generalisation of the Pascal's triangle that we named Pascal's cube, as it has the construction of a cube with entries given by extended binomial coefficients ${}^cC^{a}_{b}$. The Pascal's cube is equivalent to the well-studied Pascal's pyramid, with the advantage that the Pascal's cube can be mapped onto the Cartesian plane for easier computation. We define a stochastic process using extended binomial coefficients on the Pascal's pyramid representing the dispersion of a free particle. With some constrains, we show that this stochastic process satisfies the heat equation.