Is this sequence of polynomials well-known? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:28:48Z http://mathoverflow.net/feeds/question/45424 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45424/is-this-sequence-of-polynomials-well-known Is this sequence of polynomials well-known? David Loeffler 2010-11-09T12:29:52Z 2011-06-12T01:34:19Z <p>While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define $h_j(X)$ for $j \ge 0$ by $h_0(X) = 1$ and $$h_{j}(X) = \frac{X + 1}{j}\left(- X \frac{\mathrm{d}}{\mathrm{d}\ X} + j\right)h_{j-1}(X)$$ for $j \ge 1$.</p> <p>I was interested in these because $h_j(X)$ is the unique polynomial of degree $j$ such that $$\left(\frac{t}{e^t - 1}\right)^{j+1} \cdot h_j(e^t - 1) = 1 + O(t^{j+1}),$$ and in fact it follows from the recurrence that $$\left(\frac{t}{e^t - 1}\right)^{j+1} \cdot h_j(e^t - 1) = 1 + (-1)^j \sum_{n \ge j+1} \binom{n-1}{j} \frac{B_n t^n}{n!}$$ where $B_n$ are the usual Bernoulli numbers.</p> <p>Now, I can't believe that these polynomials $h_j$ aren't some terribly classical well-studied thing, but they don't match any of the standard sequences of polynomials I could find on the web. Does anyone recognise these?</p> http://mathoverflow.net/questions/45424/is-this-sequence-of-polynomials-well-known/45457#45457 Answer by Kevin O'Bryant for Is this sequence of polynomials well-known? Kevin O'Bryant 2010-11-09T16:36:20Z 2011-06-12T01:34:19Z <p>The first several are:</p> <p>$$0! \cdot h_0(x) = 1$$ $$1! \cdot h_1(x) = x+1$$ $$2! \cdot h_2(x) = x^2+3 x+2$$ $$3! \cdot h_3(x) = x^3+7 x^2+12 x+6$$ $$4! \cdot h_4(x) = x^4+15 x^3+50 x^2+60 x+24$$</p> <p>Feeding the sequence $2,3,1,6,12,7,1,24,60$ into the <a href="http://www.research.att.com/njas/sequences/" rel="nofollow">OEIS</a> gives the <a href="http://oeis.org/A130850" rel="nofollow">following page</a>, which contains generating functions, relations, and citations to occurrences of this sequence of polynomials in the literature.</p>