Proof step e-5648409316d57127

obreschkoff_combination → concludes about obreschkoff_combination of Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >=… JSON

Theorem

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

Operation

Obreschkoff combination Pₙ + Pₙ₊₁

{
  "operation": "obreschkoff_combination"
}

Premises

• Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and … (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
• Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and … (sequence)
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2

Claims

• { "claim": "real_rooted", "from_index": 1, "region": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: obreschkoff_combination, SHA-256 a44b56df56c1ca8723ffa3dab4ddadcd95a93c5d2a5fd5d0818a5a02b9162142, 9,441 bytes (2,506 compressed); download JSON

Cited by

• n-ac6633a57506e279 (real_rooted)

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.