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(x), Q(x) [x] are two relatively prime polynomials, and that P(x)Q(x) = n=0anxn has the property that an for all n. We show that if Q(1α) = 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