Sums Of Products Of Powers Of Trigonometric Functions On The Half-plane Averaging Angles

By means of classical analytic methods of combinatorial mathematics,such as bivariate generative functions,formal power series,and partial fractional decomposition,this dissertation investigates the problems on sums of products of powers trigonometric function with a free parameter on the half-plane averaging angles.The details are as follows:In the first chapter,we first introduce the development history and research status of the finite sum of trigonometric functions on the half-plane averaging angles,after that,we review some related concepts,symbols,conventions,formulas and methods,as well as five lemmas of trigonometric functions and formal power series that we need to use in the following paper.In the second chapter,we consider the calculation problem of sums of products of the powers of the secant and the cosecant function with a free parameter on the half-plane averaging angles,divided into finite sum and alternate sum.Since the closed summation formula for the sum of trigonometric function depend on the number of terms in the sum,which is the parity of n,four sums of trigonometric functions are considered,The bivariate generating function and the closed summation formula are given for each trigonometric sum,the summation results of sum of product with the first to fourth powers of the cosecant and the tangent function is also listed.In the third chapter,we consider the calculation problem of sums of products of the powers of the cosecant and the tangent function with a free parameter on the half-plane averaging angles,however,we discuss the sums of products of powers of the secant and the cotangent function in Chapter 4.The research method is still bivariate generative function method.We consider the finite sums and alternating sums and the parity of n,so the calculation of four sums of trigonometric functions are studied in each chapter.The summation results of related products sums in each chapter are also listed.In the last chapter,we summarize the main results of the paper.Twelve closed summation formula of the sum of trigonometric functions is listed.The transformation relation and the reverse relations among these trigonometric functions are pointed out.Several unsolved trigonometric function summation problems are also proposed.