Wikipedia:Reference desk/Archives/Mathematics/2021 February 5
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February 5
editCoefficients of a Legendre polynomial
editStarting from the definition how to show that , i.e. the coefficients of the Legendre polynomials always sum to 1? Thanks Abdul Muhsy (talk) 12:14, 5 February 2021 (UTC)
- This is equivalent to showing that the value of at equals By the substitution this is the value of at . Binomial expansion of results in a polynomial in , which, presented as a sum of terms of increasing degree, has the form The -fold derivative is then Putting we see that all but the first term vanish. --Lambiam 13:54, 5 February 2021 (UTC)
Set whose boundary is the entire space
editIs there a known name for those subsets Y of a topological space X for which one of the following four equivalent conditions is satisfied?
- Both Y and its complement are dense.
- Both Y and its complement have empty interiors.
- Y is dense and has an empty interior.
- The boundary of Y is all of X.
Any nonempty proper subset of an indiscrete space satisfies the above four equivalent conditions. Are there any examples of such subsets of the real numbers with the usual topology? GeoffreyT2000 (talk) 16:29, 5 February 2021 (UTC)
- Aren't both Q (the rational numbers) and R\Q dense in R? --Lambiam 19:36, 5 February 2021 (UTC)
- The term I've seen for the first condition is dense/co-dense. The rationals are also the example I would have given. JoelleJay (talk) 21:33, 5 February 2021 (UTC)
- In fact "dense set with empty interior" is quite common, I'd say the standard term. pma 00:06, 6 February 2021 (UTC)