On a finite sum involving inverse powers of cosines

Abstract

A finite sum Sm,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 Sm,v when v is an integer. Consequently, the first ten values of Sm,v are tabulated. © Springer Science+Busines..