DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Bernoulli polynomials by a normalized coefficient recursion

Definition

Let $ι : N \to R$ be the canonical natural map of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field. Define polynomials $B_{m} \in R [X]$ recursively. Put $B_{0} (X) = 1$ . If $m \geq 1$ and $B_{m - 1} (X) = \sum_{q = 0}^{m - 1} β_{m - 1, q} X^{q}$ , set $B_{m} (X) : = ι (m) \sum_{q = 0}^{m - 1} β_{m - 1, q} (\frac{X^{q + 1}}{ι (q + 1)} - \frac{1}{ι (q + 1) ι (q + 2)}) .$ Every sum is finite. The denominator factors are positive by Canonical naturals are positive and strictly increasing, hence invertible in $R$ by The reals form a field. Write $B_{m} : = B_{m} (0)$ for the Bernoulli number of index $m$ .

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Sources

L. Ridgway Scott, Numerical Analysis, 2nd ed., §13.3 (standard reference, not scraped)
NIST Digital Library of Mathematical Functions, Chapter 24 §24.2(i), Bernoulli Numbers and Polynomials (standard reference, not scraped)