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

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 $x^{2}-my^{2}=\pm1 \ (m\text{ is a non-square natural number})$.