On Certain Fejér and Tchebychev Type Integrals

Abstract

Let m and n denote a pair of positive integers. In this paper, we call upon the Hadamard product and computer algebra techniques to evaluate the Fejér integral π −1 ∫0 π (sin mθ / sin θ) 2n dθ. Using symmetry arguments, it is proved that the value of this integral is an odd polynomial in m of degree 2n − 1. This permits using polynomial curve fitting methods and mathematical software packages to obtain evaluation formulas for n relatively small. Some cases of the above integral with 2n replaced by 2n + 1 are also discussed. A familiar identity shows that these yield evaluations of integrals of powers of certain Tchebychev polynomials.