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 $j \ge 1$.

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.

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?

  • $\begingroup$ Is there a polynomial sequence search web thingy? I'm thinking like OEIS or ISC (inverse symbolic calculator)? $\endgroup$ – ohai Nov 9 '10 at 14:56
  • $\begingroup$ One can try putting the sequence of coefficients into the OEIS (maybe in this case normalize the coefficients so that things will be integral). $\endgroup$ – JBL Nov 9 '10 at 15:28
  • $\begingroup$ Not sure whether this is helpful: (for example using a guessing package) it's easy to see that the generating function satisfies the very nice ADE $f'(z)=f(z)^2+Xf(z)$, with explicit solution $\frac{X e^{Xz}}{1+X-e^{Xz}}$ $\endgroup$ – Martin Rubey Nov 9 '10 at 16:13

The first several are:

$$0! \cdot h_0(x) = 1$$ $$1! \cdot h_1(x) = x+1$$ $$2! \cdot h_2(x) = x^2+3 x+2$$ $$3! \cdot h_3(x) = x^3+7 x^2+12 x+6$$ $$4! \cdot h_4(x) = x^4+15 x^3+50 x^2+60 x+24$$

Feeding the sequence $2,3,1,6,12,7,1,24,60$ into the OEIS gives the following page, which contains generating functions, relations, and citations to occurrences of this sequence of polynomials in the literature.

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    $\begingroup$ You beat me :-) I just wanted to comment that the generating function above is a simple transformation of the generating function for Eulerian polynomials... $\endgroup$ – Martin Rubey Nov 9 '10 at 16:42
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    $\begingroup$ Out of interest, what made you think that that sequence would yield something? And why not e.g. 1,1,1,1,3,2,1,7,12,6,1,15? $\endgroup$ – gowers Jun 11 '11 at 7:55
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    $\begingroup$ Actually, I've just looked it up (though your second link takes me to the same page as the first) and I see that the answer is that even "unnatural" sequences such as those obtained by reading a triangle along its rows are included in OEIS, and that subsequences are recognised too. That makes it a more useful resource than I had realized. $\endgroup$ – gowers Jun 11 '11 at 7:58
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    $\begingroup$ Sometimes a given sequence "naturally" starts with a different index, so leaving off the first few terms increases the likelihood of finding the right sequence in the encyclopedia. $\endgroup$ – Kevin O'Bryant Jun 12 '11 at 1:38

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