Lesson plan
Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \{p_n(x)\}_{n=0,1,2,\ldots} satisfying the identity : \frac{d}{dx} p_n(x) = np_{n-1}(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the trivial example \{x^n\} are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.
Article summary
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \{p_n(x)\}_{n=0,1,2,\ldots} satisfying the identity : \frac{d}{dx} p_n(x) = np_{n-1}(x), and in which p_0(x) is a non-zero constant. Among the most notable Appell sequences besides the trivial example \{x^n\} are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.