#### Vol. 4, No. 2, 2011

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An observation on generating functions with an application to a sum of secant powers

### Jeffrey Mudrock

Vol. 4 (2011), No. 2, 117–125
##### Abstract

Suppose that $P\left(x\right)$, $Q\left(x\right)\in ℤ\left[x\right]$ are two relatively prime polynomials, and that $P\left(x\right)∕Q\left(x\right)={\sum }_{n=0}^{\infty }{a}_{n}{x}^{n}$ has the property that ${a}_{n}\in ℤ$ for all $n$. We show that if $Q\left(1∕\alpha \right)=0$, then $\alpha$ is an algebraic integer. Then, we show that this result can be used to provide a solution to Problem 11213(b) of the American Mathematical Monthly (2006).

##### Keywords
algebraic number theory, generating functions, secant function
Primary: 11R04
Secondary: 11R18