Cauchy product of two series

Let $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ be series in $\mathbb{R}$ or $\mathbb{C}$. Their Cauchy product is the series $\sum_{n=0}^\infty c_n$ defined by

Cauchy products correspond to multiplication of power series and to discrete convolution. Convergence of the Cauchy product requires hypotheses (e.g., absolute convergence of one factor or suitable conditions such as Mertens’ theorem).

Examples:

• If $a_n=b_n=1$ for all $n$, then $c_n=n+1$, so the Cauchy product is $\sum_{n=0}^\infty (n+1)$, which diverges.
• If $a_0=1$, $a_1=1$, $a_n=0$ for $n\ge 2$ and similarly for $b_n$, then $c_0=1$, $c_1=2$, $c_2=1$, $c_n=0$ for $n\ge 3$ (finite convolution).
• If $f(x)=\sum a_n x^n$ and $g(x)=\sum b_n x^n$, then formally $f(x)g(x)=\sum c_n x^n$ with $c_n$ as above.