On integer values of the generating functions for sequences given by the Pell's equations

TL;DR

This paper extends previous work on the integrality of generating functions for Fibonacci and Lucas sequences to those derived from solutions of more general Pell's equations, exploring how these properties change with different equations.

Contribution

It generalizes known results about generating functions' integer values from specific Pell's equations to a broader class involving non-square natural numbers.

Findings

Similar integrality results hold for sequences from Pell's equations with non-square m.

The paper characterizes when these generating functions take integer values.

It broadens understanding of the relationship between Pell's equations and generating function properties.

Abstract

D. S. Hong and P. Pongsriiam have provided a necessary and sufficient condition for the generating function for Fibonacci numbers (resp. the Lucas numbers) to be an integer value, for rational numbers. In other words, their results relate to the integer values of the generating functions of the sequences obtained from the integer solutions of Pell's equation $5x^{2}-y^{2}=\pm4$. If we change this Pell's equation to another type of Pell's equation, how will their results change? This is a natural and interesting problem. In this paper, we show that a result similar to theirs is obtained for the generating functions for sequences given by Pell's equation $x^{2}-my^{2}=\pm1 \ (m\text{ is a non-square natural number})$.