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Theorem
Let $f: (a-r, a+r) \to \R $ and $g: (a-r, a+r) \to \R $ be functions analytic on $(a-r, a+r) $, with power series expansions
$$f(x) = \sum_{ n=0 }^{ \infty } c_n(x-a)^n $$
and
$$g(x) = \sum_{ n=0 }^{ \infty }d_n(x-a)^n $$
respectively.
Then $fg: (a-r, a+r) \to \R $ is also analytic on (a-r, a+r), with power series expansion
$$f(x)g(x) = \sum_{ n=0 }^{ \infty }e_n(x-a)^n $$
where $e_n = \sum_{ m=0 }^{ n }c_md_{n-m} $
The sequence $(e_n)^ \infty_{n=0} $ is referred to as the convolution of the sequences $(c_n)^ \infty_{n=0} $ and $(d_n)^ \infty_{n=0} $.