The formal power series ring consists of all expressions with . Addition is coefficientwise, while multiplication uses the Cauchy product; no analytic convergence is required.
The ordinary generating function of a sequence is the formal power series . Algebraic identities for encode recurrences and counting constructions for the coefficients.
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Formal power series are mathematical objects used primarily in combinatorics, algebra, and related fields. A formal power series is an infinite sum of terms where each term consists of a coefficient multiplied by a variable raised to a power.