Hyperbolic Pascal triangles

Abstract

In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.

Properties of hyperbolic Pascal triangles

“…Theorem 1. (see [2]) The three sequences {a n }, {b n } and {s n } can be described by the same ternary homogeneous recurrence relation…”

“…Table 1 shows the information about the location of the terms of the binary recurrences f n = T f n−1 ± f n−2 (see [2,8]). Here, for example, LR T −1 means that going down from a given element having type A, via elements type A, first turn left, and then (T − 1)-times right.…”

“…(see [2]) The sequence {ŝ n } can be described by the ternary homogeneous recurrence relation s n = qŝ n−1 − (q + 1)ŝ n−2 + 2ŝ n−3 (n ≥ 4),…”

“…In the article [2] we introduced the so called hyperbolic Pascal triangles (in short HPT s) by extending the link between the infinite graph corresponding to the regular square Euclidean mosaic and classical Pascal's triangle to the hyperbolic plane. The purpose of this paper to make an overview about the results on HPT s and to pose some open problems.…”

Connection between bi s nomial coefficients and their analogs and symmetric functions

“…Belbachir et al [4] the hyperbolic plane. They described precisely the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic.…”