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I've been interested in proving a log-concavity result via proving that certain family of polynomials is real-rooted. By performing a sequence of transformations, I can reduce that problem to proving that the following very concrete family of polynomials are real-rooted.

$$p_n(t) = p_{n-1}(t) + \frac{t}{2} \cdot \sum_{j=2}^{n-2} \binom{n}{j}\, p_j(t)\, p_{n-j}(t).$$

With initial conditions $p_1(t)=p_2(t) = 1$. The first few values are:

$p_3(t) = 1$, $p_4(t) = 3t+1$, $p_5(t) = 13t+1$, $p_6(t) = 45t^2+38t+1$, $p_7(t) = 423t^2+94t+1$, $p_8(t) = 1575t^3+2425t^2+213t+1$.

I have tried many of the usual techniques to tackle this, without too much luck. Any suggestions are more than welcome. (It seems to be the case that the zeros of $p_n(t)$ always interlace those of $p_{n+1}(t)$, but of course I haven't been able to show it).

EDIT: The bounty would also be awarded if someone can prove the log-concavity of the coefficients of each $p_n(t)$ (independently of the real-rootedness).

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    $\begingroup$ For what it worth, the generating function $\sum_{n=2}^\infty p_n/n! x^n$ satisfies an ode $(f-t f^2/2)'=2x+f$. $\endgroup$ Commented Oct 25, 2023 at 18:35
  • $\begingroup$ I cannot currently provide a full answer but I believe that Lemmas 3.14 and 3.16 in Fisk's book about interlacing should be enough to prove this with a bit of effort. In particular, as you correctly point out the key might be interlacing. Once interlacing is proved a modified application of Lemma 3.16 does the job. Proving interlacing is a bit trickier, but I believe that the fact that the polynomials have all positive coefficients allows for an application of a variant of Lemma 3.14 for variable alphas. Did you try this? $\endgroup$ Commented Oct 29, 2023 at 12:36
  • $\begingroup$ @Hvjurthuk I don't see how you plan to use these two lemmas precisely (or perhaps I'm looking at the wrong version of Fisk's book). Can you elaborate a bit more? $\endgroup$ Commented Oct 29, 2023 at 16:14
  • $\begingroup$ @FedorPetrov I think the right hand side should be $x+f$. In fact $f$ fulfills a messy transcendental equation (without derivatives of $f$), which becomes somewhat shorter if we write it in terms of $g=(tf-1)/(1+tx)$. But I doubt that it helps in any way. $\endgroup$ Commented Oct 29, 2023 at 22:56
  • $\begingroup$ Just to mark a dead end, I'll note that neither the leading coefficients or the values of $p_n(1)$ seem to appear in OEIS. $\endgroup$ Commented Mar 29, 2024 at 21:23

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The sequences $\{p_n\}$ or $\{p_{3n}\}$ do not seem to satisfy nice linear recursions involving derivatives (similar to how Eulerian polynomials can be shown to be real-rooted).

If you have a combinatorial interpretation for the coefficients, then perhaps you can refine your family, so that you get more polynomials but easier recursions, and prove the interlacing property that way.

See this page for some references on various techniques to prove real-rootedness.

A long shot is: perhaps your polynomials are matching polynomials, (or independence set polynomials) of some graphs, and then real-rootedness follows.

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  • $\begingroup$ Thanks Per. I do not have a combinatorial interpretation for these coefficients, but rather of a (too complicated) transformation of them. I am aware of the techniques to prove the real-rootedness for the Eulerian polynomials and other similar sequences, and I agree that those techniques do not seem to apply here. That is, in a sense, the reason I asked this question. By the way, shouldn't this "answer" be a "comment"? $\endgroup$ Commented Oct 26, 2023 at 9:33
  • $\begingroup$ @LuisFerroni Yes, sure, but its too long for a comment. In any case, a refinement would perhaps help. For example, Eulerian polynomials A_n, but where one sum over permutations where n is sent to k, (as k varies) gives a refinement which simplifies the proof of real-rootedness, see Petter Bränden's chapter on Log-concavity, realrootedness and beyond. $\endgroup$ Commented Oct 26, 2023 at 21:01

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