[Paper Review] Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers

Martin Erik Horn|ArXiv.org|Nov 26, 2007

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper generalizes the classical Pascal triangle into multi-dimensional Pauli Pascal hyperpyramids using anti-commutative Pauli matrices, extending Fibonacci and Jacobsthal sequences into Pauli Fibonacci and Pauli Jacobsthal numbers of higher order. The work introduces algebraic structures based on Pauli algebra to generate recursive number sequences with non-commutative properties, offering a novel combinatorial framework in non-associative algebraic systems.

ABSTRACT

The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli Jacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits?

Motivation & Objective

• To extend the classical Pascal triangle into higher-dimensional hyperpyramids using Pauli matrices.
• To generalize Fibonacci and Jacobsthal number sequences into non-commutative, Pauli-based analogues.
• To explore recursive number sequences defined through anti-commutative algebraic relations.
• To investigate the structural and combinatorial properties of these generalized sequences in multi-dimensional settings.

Proposed method

• Construction of three anti-commutative two-dimensional Pauli Pascal triangles using Pauli matrices σ_x, σ_y, σ_z.
• Generalization of these triangles into multi-dimensional Pauli Pascal hyperpyramids via tensor products and recursive addition rules.
• Definition of Pauli Fibonacci and Pauli Jacobsthal numbers using recursive relations based on Pauli matrix algebra.
• Derivation of higher-order Pauli Fibonacci sequences through iterative application of non-commutative recurrence relations.
• Use of matrix-valued coefficients to encode combinatorial coefficients in the hyperpyramids.
• Application of anti-commutation relations {σ_i, σ_j} = 2δ_ij I to ensure non-commutative structure in the number sequences.

Experimental results

Research questions

• RQ1 How can the Pascal triangle be generalized into multi-dimensional hyperpyramids using Pauli matrices?
• RQ2 What are the properties of Fibonacci and Jacobsthal sequences when extended into non-commutative, Pauli-based algebraic systems?
• RQ3 Can higher-order recursive sequences be defined using Pauli matrix algebra in a hyperpyramid framework?
• RQ4 What algebraic constraints emerge from anti-commutativity in these generalized number systems?
• RQ5 Do these generalized sequences exhibit structural or combinatorial patterns analogous to classical sequences?

Key findings

• The Pauli Pascal hyperpyramid generalizes the classical Pascal triangle into multi-dimensional structures using anti-commutative Pauli matrices.
• Pauli Fibonacci and Pauli Jacobsthal numbers are defined through recursive relations involving Pauli matrix algebra, preserving sequence-like behavior under non-commutative rules.
• Higher-order Pauli Fibonacci sequences are constructed via iterative application of non-commutative recurrence relations in multi-dimensional lattices.
• The hyperpyramid framework supports three distinct anti-commutative triangles, each corresponding to a different Pauli matrix pair.
• The paper establishes a formal link between combinatorial number sequences and non-associative algebraic systems through Pauli matrix representations.
• The work poses the rhetorical question 'are Pauli rabbits killer rabbits?' to highlight the non-intuitive, non-commutative nature of the sequences.