Proof step e-b96b8d87f5e7b835

obreschkoff_combination → concludes about obreschkoff_combination of Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1. 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_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
• Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1. (sequence)
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1

Claims

• { "claim": "real_rooted", "from_index": 0, "region": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: obreschkoff_combination, SHA-256 adfa2219d713af9c0a2fb497d5fe1f5becebfd9c8338a847cc875488a7be75b7, 28,853 bytes (4,935 compressed); download JSON

Cited by

• n-c94fb4ea603ebbf0 (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.