Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
Is there a polynomial sequence search web thingy? I'm thinking like OEIS or ISC (inverse symbolic calculator)? –  ohai Nov 9 '10 at 14:56
    
One can try putting the sequence of coefficients into the OEIS (maybe in this case normalize the coefficients so that things will be integral). –  JBL Nov 9 '10 at 15:28
    
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 $\frac{X e^{Xz}}{1+X-e^{Xz}}$ –  Martin Rubey Nov 9 '10 at 16:13

1 Answer 1

up vote 14 down vote accepted

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.

share|improve this answer
2  
You beat me :-) I just wanted to comment that the generating function above is a simple transformation of the generating function for Eulerian polynomials... –  Martin Rubey Nov 9 '10 at 16:42
1  
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? –  gowers Jun 11 '11 at 7:55
4  
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. –  gowers Jun 11 '11 at 7:58
2  
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. –  Kevin O'Bryant Jun 12 '11 at 1:38

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.