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

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