Alternating sums in hyperbolic Pascal triangles

TL;DR

This paper generalizes the calculation of alternating sums in hyperbolic Pascal triangles, extending previous results from a specific case to a broader class associated with regular mosaics in the hyperbolic plane.

Contribution

It provides a general formula for the alternating sums in hyperbolic Pascal triangles for all regular mosaics of the form {4,q} with q ≥ 5.

Findings

Derived a general expression for alternating sums in {4,q} hyperbolic Pascal triangles.

Extended previous specific case results to a broader class of hyperbolic mosaics.

Enhanced understanding of combinatorial properties in hyperbolic geometric structures.

Abstract

A new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced in [H.B, L.N, L.Sz: Hyperbolic Pascal triangles]. The mathematical background goes back to the regular mosaics in the hyperbolic plane. The alternating sum of elements in the rows was given in the special case $\{4,5\}$ of the hyperbolic Pascal triangles. In this article, we determine the alternating sum generally in the hyperbolic Pascal triangle corresponding to $\{4,q\}$ with $q\ge5$.