= D-finite power series
{c}
= D-finite
{synonym}
A <formal power series> $f(x)$ over a <field> of <characteristic> zero is <D-finite> if it satisfies a nontrivial linear <differential equation> with <polynomial coefficients>:
$$
p_0(x)f(x)+p_1(x)f'(x)+\cdots+p_r(x)f^{(r)}(x)=0.
$$
The <polynomials> are not all zero. Equivalently, the coefficients of $f$ satisfy a linear <recurrence relation> with <polynomial> functions of the index as coefficients.
For example, $e^x$ satisfies $f'-f=0$, and every <rational function> regular at zero is <D-finite>. The class is closed under sums, products, and <derivatives>. This permits symbolic computation with many <generating functions> using finite equations rather than an infinite list of coefficients.
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