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

Mudrock, Jeffrey

doi:10.2140/involve.2011.4.117

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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

DOI: 10.2140/involve.2011.4.117

Abstract

Suppose that $P (x)$ , $Q (x) \in ℤ [x]$ are two relatively prime polynomials, and that $P (x) ∕ Q (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}$ has the property that $a_{n} \in ℤ$ for all $n$ . We show that if $Q (1 ∕ α) = 0$ , then $α$ 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

Mathematical Subject Classification 2000

Primary: 11R04

Secondary: 11R18

Milestones

Received: 19 July 2010

Revised: 1 February 2011

Accepted: 2 February 2011

Published: 17 January 2012

Communicated by Nigel Boston

Authors

Jeffrey Mudrock
Mathematics Department University of Illinois at Urbana-Champaign 1409 West Green Street Urbana, IL 61801 United States

Vol. 4, No. 2, 2011