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

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.