Pascal Pyramids, Pascal Hyper-Pyramids and a Bilateral Multinomial Theorem

TL;DR

This paper generalizes Pascal's triangle into higher-dimensional pyramids and extends the bilateral binomial theorem into bilateral multinomial theorems, broadening combinatorial and algebraic frameworks.

Contribution

It introduces higher-dimensional Pascal pyramids and hyper-pyramids and generalizes the bilateral binomial theorem to bilateral multinomial theorems, expanding mathematical structures.

Findings

Definition of Pascal pyramids and hyper-pyramids in multiple dimensions

Formulation of bilateral multinomial theorem

Potential applications in combinatorics and algebra

Abstract

Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral Trinomial Theorem resp. a Bilateral Multinomial Theorem.