Symbolic computation of some power-trigonometric series

Let f∗(z) = ∞ ∑ j=0 aj z j be a convergent series in which {aj}j=0 are known real numbers. In this paper, by referring to Osler’s lemma [8], we obtain explicit forms of the two bivariate series ∞ ∑ j=0 an j+m r j cos(α+ j)θ and ∞ ∑ j=0 an j+m r j sin(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.