Symbolic computation of some power-trigonometric series

Let f⁎(z)=∑j=0∞aj⁎zj be a convergent series in which {aj⁎}j=0∞ are known real numbers. In this paper, by referring to Osler's lemma (Osler, 1975), we obtain explicit forms of the two bivariate series∑j=0∞anj+m⁎rjcos⁡(α+j)θand∑j=0∞anj+m⁎rjsin⁡(α+j)θ, where r, θ are real variables, α∈R, n∈N and m∈{0,1,…,n−1}. With some illustrative examples, we also show how to obtain the explicit form of a trigonometric series when f⁎(z) is explicitly given. Three new integral formulae are derived in this direction.