Alternating sums in hyperbolic Pascal triangles

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 $q\ge5$.