Proof step `e-9acfc4c7c0e80a75`

preserver_image derivative_polynomial, shift 0 → concludes about Triangle read by rows: A008277 * A007318. 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": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Premises

Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished. (sequence)
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "0" } } }

Verification and certificate

polytool: replayed (the certificate replayed when the step was written)
Lean: not attempted
certificate: preserver_image, SHA-256 42cd4f96bfe6aeb28cbbe7ee5e75cac263e3bb8e36c15f5cd9e5829b6be45fac, 8,698 bytes (2,490 compressed); download JSON

Cited by

n-bdae5aa3d1cac0ec (one_signed_coefficients)
n-bdae5aa3d1cac0ec (hadamard_kernel)
n-bdae5aa3d1cac0ec (real_rooted)

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-9acfc4c7c0e80a75