Proof step `e-2ca40a069e1a39d6`

preserver_image derivative_polynomial, shift 2 → concludes about Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018). JSON

Theorem

Laguerre’s theorem on differential operators [SP14] polytool ?

Lean (RealRooted): RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Operation

Image under a real-rootedness preserver

{
  "index_shift": 2,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2. (sequence)
uses: Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]

Claims

{ "claim": "real_rooted", "from_index": 1, "region": { "upper": { "closed": true, "value": "0" } } }
{ "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "unordered" }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 18c4868e4e8446adf74ff1bd245bdbca4b93d0495097d13a2696852a9ca0f6be, 13,680 bytes (3,415 compressed); download JSON

Cited by

n-1e9cff7ed448d4e0 (real_rooted)
n-1e9cff7ed448d4e0 (one_signed_coefficients)
n-1e9cff7ed448d4e0 (hadamard_kernel)
n-1e9cff7ed448d4e0 (interlacing)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.

Proof step e-2ca40a069e1a39d6