Symbolic computation of some power-trigonometric series. (May 2017) - Masjed-Jamei, Mohammad; Koepf, Wolfram
Abstract: Let f ⁎ ( z ) = ∑ j = 0 ∞ a j ⁎ z j be a convergent series in which a j ⁎ 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 ∞ a n j + m ⁎ r j cos ⁡ ( α + j ) θ and ∑ j = 0 ∞ a n 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.