## Abstract

While counting lattice points in octahedra of different dimensions n and m, it is an interesting question to ask, how many octahedra exist containing equally many such points. This gives rise to the Diophantine equation Pn(x) = P _{m}(y) in rational integers x,y, where {P _{k}(x)} denote special Meixner polynomials {M _{k} ^{(β}, ^{c)}(x)} with β = 1, c = -1. We join the purely algebraic criterion of Y. Bilu and R. F. Tichy (The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), no. 3, 261-288) with a famous result of P. Erdös and J. L. Selfridge (The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), 292-301) and prove that (euqation presented) with m, n ≥ 3, β ∈ ℤ/{0,-1,-2,-max(n,m) + 1} and c1,c _{2} ∈ ℚ \ {0, 1} only admits a finite number of integral solutions x, y. Some more results on polynomial families in three-term recurrences are presented.

Original language | English |
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Pages | 724-729 |

Number of pages | 6 |

Publication status | Published - 1 Dec 2006 |

Event | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, United States Duration: 19 Jun 2006 → 23 Jun 2006 |

### Conference

Conference | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 |
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Country/Territory | United States |

City | San Diego |

Period | 19/06/06 → 23/06/06 |

## Keywords

- Counting lattice points
- Diophantine equations
- Meixner polynomials
- Orthogonal polynomials

## ASJC Scopus subject areas

- Algebra and Number Theory