Testwiki:Reference desk/Archives/Mathematics/2021 February 5

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < February 4 ! width="25%" align="center"|<< Jan | February | Mar >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

February 5

Coefficients of a Legendre polynomial

Starting from the definition $P_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}(x^2-1{)}^n$ how to show that $P_n(1)=1$, 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 $\frac{d^n}{d{x}^n}(x^2-1{)}^n$ at $x=1$ equals $2^{n}n!.$ By the substitution $x=z+1,$ this is the value of $\frac{d^n}{d{z}^n}(2z+z^2{)}^n$ at $z=0$. Binomial expansion of $(2z+z^2{)}^n$ results in a polynomial in $z$, which, presented as a sum of terms of increasing degree, has the form $2^n{z}^n+c_{n+1}{z}^{n+1}+....$ The $n$-fold derivative is then $2^{n}n!+c_{n+1}(n+1)!z+....$ Putting $z=0,$ we see that all but the first term vanish.  --Lambiam 13:54, 5 February 2021 (UTC)

Set whose boundary is the entire space

Is there a known name for those subsets Y of a topological space X for which one of the following four equivalent conditions is satisfied?

1. Both Y and its complement are dense.
2. Both Y and its complement have empty interiors.
3. Y is dense and has an empty interior.
4. 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 $ℝ$ 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)