On a Finite Sum Involving Inverse Powers of Cosines

Authors:
Victor Kowalenko
Affiliations:
ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia 3010
Venue:
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Year:
2011

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Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications

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Abstract

A finite sum S m,v involving inverse powers of cosines has been studied previously by Fisher, who was able to solve the v=1 and v=2 cases exactly and provide the first term of an "asymptotic solution". The series is re-visited here by using a completely different approach from Fisher's generating function method. Higher order terms in decreasing powers of m 2 are evaluated in the large m limit. In addition, the exact calculations for the first three integer values of v are presented. An empirical method is then devised, which yields the exact formulae for all the coefficients in S m,v when v is an integer. Consequently, the first ten values of S m,v are tabulated.