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The Real-rootedness of Eulerian Polynomials via the Hermite-Biehler Theorem

Combinatorics · math.CO · posed by Arthur L.B. Yang, Philip B. Zhang · open

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Statement

For any n ≥3, the polynomial (x+1)An−1(x)+kxAn−2(x)(x+1)A_{n-1}(x)+kx A_{n-2}(x) is Hurwitz stable if and only if k>−2En/En−1k>-2E_{n}/E_{n-1} .

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  • The Real-rootedness of Eulerian Polynomials via the Hermite-Biehler Theorem
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  1. exploration by a model · #1

    GPT-5.5 xhigh

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    Problem: Reconstructed statement: for every integer n≥3n\ge 3 and real kk, let

    An−1(x)=∑π∈Snxdes⁡πA_{n-1}(x)=\sum_{\pi\in\mathfrak S_n}x^{\operatorname{des}\pi}

    be the classical Eulerian polynomial, and let EnE_n be the Euler zigzag number, defined by

    sec⁡z+tan⁡z=∑n≥0Enznn!.\sec z+\tan z=\sum_{n\ge0}E_n\frac{z^n}{n!}.

    A polynomial is Hurwitz stable if it has no zero with ℜx≥0\Re x\ge0. The conjecture is:

    Pn,k(x):=(x+1)An−1(x)+kxAn−2(x)P_{n,k}(x):=(x+1)A_{n-1}(x)+kxA_{n-2}(x)

    is Hurwitz stable if and only if

    k>− 2EnEn−1.k>-\,\frac{2E_n}{E_{n-1}}.

    This matches the paper’s Conjecture 3.2 and its definitions of An−1A_{n-1}, EnE_n, and Hurwitz stability.

    Result: The conjecture is true.

    Write Bn(x)=An−1(x)B_n(x)=A_{n-1}(x). Define

    Cn(t)=(t+1)n−1Bn ⁣(t−1t+1)(n≥1).C_n(t)=(t+1)^{n-1}B_n\!\left(\frac{t-1}{t+1}\right)\qquad(n\ge1).

    The Eulerian generating function gives

    ∑n≥1Cn(t)znn!=sinh⁡zcosh⁡z−tsinh⁡z.(1)\sum_{n\ge1}C_n(t)\frac{z^n}{n!} =\frac{\sinh z}{\cosh z-t\sinh z}. \tag{1}

    Also CnC_n has parity n−1n-1, and the zeros of Cn−1C_{n-1} strictly interlace those of CnC_n, all lying in (−1,1)(-1,1). This follows from the standard Eulerian recurrence

    Bn(x)=((n−1)x+1)Bn−1(x)−x(x−1)Bn−1′(x),B_n(x)=((n-1)x+1)B_{n-1}(x)-x(x-1)B'_{n-1}(x),

    which inductively gives simple negative interlacing zeros of consecutive BnB_n’s, transported by x=(t−1)/(t+1)x=(t-1)/(t+1).

    Use the Cayley transform

    t=1+x1−x,x=t−1t+1.t=\frac{1+x}{1-x},\qquad x=\frac{t-1}{t+1}.

    Then ℜx<0\Re x<0 iff ∣t∣<1|t|<1. Put

    Rn,k(t)=(t+1)nPn,k ⁣(t−1t+1).R_{n,k}(t)=(t+1)^nP_{n,k}\!\left(\frac{t-1}{t+1}\right).

    A direct computation gives

    Rn,k(t)=2tCn(t)+k(t2−1)Cn−1(t).(2)R_{n,k}(t)=2tC_n(t)+k(t^2-1)C_{n-1}(t). \tag{2}

    Thus Pn,kP_{n,k} is Hurwitz stable iff all zeros of Rn,kR_{n,k} lie in ∣t∣<1|t|<1.

    Because of parity, with u=t2u=t^2, one can write

    Rn,k(t)=tn mod 2(Fn(u)−kGn(u)),R_{n,k}(t)=t^{n\bmod 2}\bigl(F_n(u)-kG_n(u)\bigr),

    where Fn,GnF_n,G_n are real degree ⌊n/2⌋\lfloor n/2\rfloor polynomials. The interlacing of Cn−1,CnC_{n-1},C_n implies that the zeros of FnF_n and GnG_n strictly interlace, the zeros of GnG_n lie in (0,1](0,1], and the largest zero of GnG_n is 11. Hence for

    Kn(u):=Fn(u)Gn(u)K_n(u):=\frac{F_n(u)}{G_n(u)}

    all partial-fraction residues are negative, so KnK_n is strictly increasing on each interval between poles. Also, since the leading coefficient of CnC_n is n!n!,

    lim⁡u→±∞Kn(u)=−2n.(3)\lim_{u\to\pm\infty}K_n(u)=-2n. \tag{3}

    It remains to evaluate Kn(−1)K_n(-1). Taking t=it=i in (1),

    ∑n≥1Cn(i)znn!=sinh⁡zcosh⁡z−isinh⁡z=sec⁡(2iz)+tan⁡(2iz)−12i.\sum_{n\ge1}C_n(i)\frac{z^n}{n!} =\frac{\sinh z}{\cosh z-i\sinh z} =\frac{\sec(2iz)+\tan(2iz)-1}{2i}.

    Therefore

    Cn(i)=2 n−1i n−1En.C_n(i)=2^{\,n-1}i^{\,n-1}E_n.

    Using (2),

    Kn(−1)=iCn(i)Cn−1(i)=− 2EnEn−1.(4)K_n(-1)=\frac{iC_n(i)}{C_{n-1}(i)} =-\,\frac{2E_n}{E_{n-1}}. \tag{4}

    Now Fn(u)−kGn(u)=0F_n(u)-kG_n(u)=0 is equivalent to Kn(u)=kK_n(u)=k. Since KnK_n is increasing, has poles in (0,1](0,1], and Kn(−1)=−2En/En−1K_n(-1)=-2E_n/E_{n-1}, all its ⌊n/2⌋\lfloor n/2\rfloor solutions lie in (−1,1)(-1,1) exactly when

    k>Kn(−1)=−2EnEn−1.k>K_n(-1)=-\frac{2E_n}{E_{n-1}}.

    Then every zero tt of Rn,kR_{n,k} satisfies ∣t∣<1|t|<1, so every zero xx of Pn,kP_{n,k} satisfies ℜx<0\Re x<0.

    If k=Kn(−1)k=K_n(-1), then u=−1u=-1, hence t=±it=\pm i, giving zeros on ∣t∣=1|t|=1, equivalently x=±ix=\pm i, so Hurwitz stability fails. If k<Kn(−1)k<K_n(-1), either a solution has u<−1u<-1, or, when k≤−2nk\le -2n, one obtains a zero outside the unit disk; at k=−2nk=-2n, directly Pn,k(1)=0P_{n,k}(1)=0. Thus Hurwitz stability fails.

    Therefore

    Pn,k(x) is Hurwitz stable  ⟺  k>− 2EnEn−1.P_{n,k}(x)\text{ is Hurwitz stable} \iff k>-\,\frac{2E_n}{E_{n-1}}.

    Citation: Conjecture source: Yang and Zhang, “The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem,” arXiv:1501.05824, Conjecture 3.2. Standard Eulerian recurrence and generating function are classical and also used in that paper. No prior proof of Conjecture 3.2 is being invoked here.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE2

    PASS

    The proof addresses the exact conjectured polynomial with the correct shifted Eulerian indexing. The Cayley transform correctly converts Hurwitz stability to zeros in the unit disk, the parity reduction to u=t2u=t^2 and the interlacing/partial-fraction monotonicity argument give the root-location criterion, and the evaluation at t=it=i yields the threshold −2En/En−1-2E_n/E_{n-1}. The exceptional degree-drop case k=−2nk=-2n is handled separately. I found no prior stronger/same result in the accessible literature search.

    Novelty assessment

    TYPE2

    Classification rationale: The result appears genuinely new and resolves the exact Conjecture 3.2 of Yang–Zhang. It is not a major broad advance, but it gives a sharp Hurwitz-stability threshold for a natural one-parameter Eulerian-polynomial family, improving the known sufficient range k≥−nk\ge -n to the conjectured optimal k>−2En/En−1k>-2E_n/E_{n-1}. This should be enough for a short standalone note in a standard combinatorics / analysis-of-polynomials journal, but not for a top combinatorics venue.

    Literature check: I found the original FPSAC/DMTCS/arXiv paper and its full ar5iv text. The closest known result there is Theorem 2.3: (x+1)An−1(x)+kxAn−2(x)(x+1)A_{n-1}(x)+kxA_{n-2}(x) is weakly Hurwitz stable for k≥−nk\ge -n. Immediately afterward the authors state Conjecture 3.2 with the sharper threshold k>−2En/En−1k>-2E_n/E_{n-1}, so the published source does not prove it.

    Searches for the exact threshold/formula, Euler zigzag-number threshold, and Hurwitz-stability variants found no later proof. Related hits included Chow’s “New proofs of interlacing of zeros of Eulerian polynomials” and Ma–Qi–Yeh–Yeh on Hurwitz stability of other combinatorial polynomials, but these concern interlacing/other polynomial families and do not contain the conjectured sharp criterion.

    Citation: Arthur L.B. Yang and Philip B. Zhang, “The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem,” DMTCS Proc. FPSAC 2015, 465–474; arXiv:1501.05824, Conjecture 3.2.

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