If I know generating funcions for sequences $$A: a_0, a_1, a_2, a_3, a_4, \dots$$ and $$B: b_0, b_1, b_2, b_3, b_4, \dots$$ and I want to find a new generating function for $$C: a_0b_0, a_1b_1, a_2b_2, a_3b_3, a_4b_4,\dots$$ How would I go about doing this? I would like an explanation, thanks!
1 Answer
Alf van der Poorten, wrote about this, calling it the "Hadamard product" of the two generating series. It's not easy stuff, but you can read about it in section 8 of this paper.
See the example in Section 11, where $$\sum{2h\choose h}x^h=(1-4x)^{-1/2}$$ is an elementary and algebraic function, but $$\sum{2h\choose h}^2x^h$$ is a complete elliptic integral (so not elementary and not algebraic).
EDIT: The link I gave above no longer works. A workaround is to type ""power series representing algebraic functions" into Google. Several links to work of Alan Baker come up, but also one to ibrarian.net, which will download a pdf of Alf's paper. Perhaps this link will work.
MORE EDIT: I should have included the bibliographic details for Alf's paper:
A. J. van der Poorten. Power series representing algebraic functions. Seminaire de Theorie des Nombres, Paris, 1990–91, pp. 241–262. Birkhauser Boston, 1993, download in pdf, review by Paul M. Eakin.